{{Short description|Infinite sum}}
{{About|infinite sums|finite sums|Summation}}
{{Calculus}}

In [[mathematics]], a '''series''' is, roughly speaking, an [[addition]] of [[Infinity|infinitely]] many [[Addend|terms]], one after the other.<ref>{{cite book |title=Calculus Made Easy |last1=Thompson |first1=Silvanus |author-link1=Silvanus P. Thompson |last2=Gardner |first2=Martin |author-link2=Martin Gardner |year=1998 |publisher=Macmillan |isbn=978-0-312-18548-0 |url=https://archive.org/details/calculusmadeeasy00thom_0 }}</ref> The study of series is a major part of [[calculus]] and its generalization, [[mathematical analysis]]. Series are used in most areas of mathematics, even for studying finite structures in [[combinatorics]] through [[generating function]]s. The mathematical properties of infinite series make them widely applicable in other quantitative disciplines such as [[physics]], [[computer science]], [[statistics]] and [[finance]].

Among the [[Ancient Greece|Ancient Greeks]], the idea that a [[potential infinity|potentially infinite]] [[summation]] could produce a finite result was considered [[paradox]]ical, most famously in [[Zeno's paradoxes]].<ref name=":1">{{Citation |last=Huggett |first=Nick |title=Zeno's Paradoxes |year=2024 |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/archives/spr2024/entries/paradox-zeno/ |access-date=2024-03-25 |edition=Spring 2024 |publisher=Metaphysics Research Lab, Stanford University |editor2-last=Nodelman |editor2-first=Uri |encyclopedia=The Stanford Encyclopedia of Philosophy}}</ref><ref>{{harvnb|Apostol|1967|pp=374–375}}</ref> Nonetheless, infinite series were applied practically by Ancient Greek mathematicians including [[Archimedes]], for instance in the [[Quadrature of the Parabola|quadrature of the parabola]].<ref>{{Cite journal |last1=Swain |first1=Gordon |last2=Dence |first2=Thomas |year=1998 |title=Archimedes' Quadrature of the Parabola Revisited |url=https://www.jstor.org/stable/2691014 |journal=Mathematics Magazine |volume=71 |issue=2 |pages=123–130 |doi=10.2307/2691014 |issn=0025-570X |jstor=2691014|url-access=subscription }}</ref><ref name=":6">{{Cite book |last=Russo |first=Lucio |title=The Forgotten Revolution |year=2004 |publisher=Springer-Verlag |isbn=978-3-540-20396-4 |location=Germany |pages=49–52 |translator-last=Levy |translator-first=Silvio}}</ref> The mathematical side of Zeno's paradoxes was resolved using the concept of a [[limit (mathematics)|limit]] during the 17th century, especially through the early calculus of [[Isaac Newton]].<ref>{{harvnb|Apostol|1967|p=377}}</ref> The resolution was made more rigorous and further improved in the 19th century through the work of [[Carl Friedrich Gauss]] and [[Augustin-Louis Cauchy]],<ref>{{harvnb|Apostol|1967|p=378}}</ref> among others, answering questions about which of these sums exist via the [[completeness of the real numbers]] and whether series terms can be rearranged or not without changing their sums using [[absolute convergence]] and [[conditional convergence]] of series. 

In modern terminology, any ordered [[Sequence (mathematics)|infinite sequence]] <math>(a_1,a_2,a_3,\ldots)</math> of terms, whether those terms are numbers, [[Function (mathematics)|functions]], [[Matrix (mathematics)|matrices]], or anything else that can be added, defines a series, which is the addition of the {{tmath|a_i}} one after the other. To emphasize that there are an infinite number of terms, series are often also called '''infinite series''' to contrast with [[finite series]], a term sometimes used for [[summation|finite sums]]. Series are represented by an [[expression (mathematics)|expression]] like
<math display=block>a_1+a_2+a_3+\cdots,</math>
or, using [[capital-sigma notation|capital-sigma summation notation]],<ref name=":5">{{harvnb|Apostol|1967|p=37}}</ref>
<math display=block>\sum_{i=1}^\infty a_i.</math>

The infinite sequence of additions expressed by a series cannot be explicitly performed in sequence in a finite amount of time. However, if the terms and their finite sums belong to a [[Set (mathematics)|set]] that has [[Limit (mathematics)|limits]], it may be possible to assign a value to a series, called the '''sum of the series'''. This value is the limit as {{tmath|n}} tends to [[infinity]] of the finite sums of the {{tmath|n}} first terms of the series if the limit exists.<ref name=":4">{{harvnb|Spivak|2008|pp=471–472}}</ref><ref name=":2">{{harvnb|Apostol|1967|p=384}}</ref><ref name=":3">{{Cite book |last1=Ablowitz |first1=Mark J. |title=Complex Variables: Introduction and Applications |last2=Fokas |first2=Athanassios S. |publisher=Cambridge University Press |year=2003 |isbn=978-0-521-53429-1 |edition=2nd |pages=110}}</ref> These finite sums are called the '''{{vanchor|partial sum}}s''' of the series. Using summation notation,
<math display=block>\sum_{i=1}^\infty a_i  = \lim_{n\to\infty}\, \sum_{i=1}^n a_i,</math>
if it exists.<ref name=":4" /><ref name=":2" /><ref name=":3" /> When the limit exists, the series is '''convergent''' or '''summable''' and also the sequence <math>(a_1,a_2,a_3,\ldots)</math> is '''summable''', and otherwise, when the limit does not exist, the series is '''divergent'''.<ref name=":4" /><ref name=":2" /><ref name=":3" />

The expression <math display=inline>\sum_{i=1}^\infty a_i</math> denotes both the series—the implicit process of adding the terms one after the other indefinitely—and, if the series is convergent, the sum of the series—the explicit limit of the process. This is a generalization of the similar convention of denoting by <math>a+b</math> both the [[addition]]—the process of adding—and its result—the ''sum'' of {{tmath|a}} and {{tmath|b}}.

Commonly, the terms of a series come from a [[Ring (mathematics)|ring]], often the [[Field (mathematics)|field]] <math>\mathbb R</math> of the [[real number]]s or the field <math>\mathbb C</math> of the [[complex number]]s. If so, the set of all series is also itself a ring, one in which the addition consists of adding series terms together term by term and the multiplication is the [[Cauchy product]].<ref name=":7">{{Cite book |last1=Dummit |first1=David S. |title=Abstract Algebra |last2=Foote |first2=Richard M. |publisher=John Wiley and Sons |year=2004 |isbn=978-0-471-43334-7 |edition=3rd |location=Hoboken, NJ |pages=238}}</ref><ref name=":422">{{harvnb|Spivak|2008|pp=486–487, 493}}</ref><ref name=":8">{{Cite book |last=Wilf |first=Herbert S. |title=Generatingfunctionology |publisher=Academic Press |year=1990 |isbn=978-1-48-324857-8 |location=San Diego |pages=27–28}}</ref>

==Definition==

=== Series ===
A ''series'' or, redundantly, an ''infinite series'', is an infinite sum. It is often represented as<ref name=":5" /><ref>{{Cite book |last=Swokoski |first=Earl W. |title=Calculus with Analytic Geometry |publisher=Prindle, Weber & Schmidt |year=1983 |isbn=978-0-87150-341-1 |edition=Alternate |location=Boston |pages=501}}</ref><ref name=":15">{{harvnb|Rudin|1976|p=59}}</ref>
<math display=block>a_0 + a_1 + a_2 + \cdots \quad \text{or} \quad a_1 + a_2 + a_3 + \cdots, </math>
where the [[summand|terms]] <math>a_k</math> are the members of a [[sequence]] of  [[number]]s, [[Function (mathematics)|functions]], or anything else that can be [[addition|added]]. A series may also be represented with [[capital-sigma notation]]:<ref name=":5" /><ref name=":15" />
<math display=block>\sum_{k=0}^{\infty} a_k \qquad \text{or} \qquad \sum_{k=1}^{\infty} a_k . </math>

It is also common to express series using a few first terms, an ellipsis, a general term, and then a final ellipsis, the general term being an expression of the {{tmath|n}}th term as a [[function (mathematics)|function]] of {{tmath|n}}:
<math display=block>a_0 + a_1 + a_2 + \cdots + a_n +\cdots \quad \text{ or } \quad f(0) + f(1) + f(2) + \cdots + f(n) + \cdots. </math>
For example, [[Euler's number]] can be defined with the series
<math display=block>\sum_{n=0}^\infty \frac 1{n!}=1+1+\frac12 +\frac 16 +\cdots + \frac 1{n!}+\cdots, </math> 
where <math>n!</math> denotes the product of the <math>n</math> first [[positive integer]]s, and <math>0!</math> is conventionally equal to <math>1.</math><ref name=":42">{{harvnb|Spivak|2008|p=426}}</ref><ref name=":22">{{harvnb|Apostol|1967|p=281}}</ref><ref>{{harvnb|Rudin|1976|p=63}}</ref>

=== Partial sum of a series ===
Given a series <math display=inline>s=\sum_{k=0}^\infty a_k</math>, its {{tmath|n}}th ''partial sum'' is<ref name=":4" /><ref name=":2" /><ref name=":3" /><ref name=":15" />
<math display=block>s_n = \sum_{k=0}^{n} a_k = a_0 + a_1 + \cdots + a_n .</math>

Some authors directly identify a series with its sequence of partial sums.<ref name=":4" /><ref name=":3" /> Either the sequence of partial sums or the sequence of terms completely characterizes the series, and the sequence of terms can be recovered from the sequence of partial sums by taking the differences between consecutive elements,
<math display=block>a_n = s_{n} - s_{n-1}. </math>

Partial summation of a sequence is an example of a linear [[sequence transformation]], and it is also known as the [[prefix sum]] in [[computer science]]. The inverse transformation for recovering a sequence from its partial sums is the [[finite difference]], another linear sequence transformation. 

Partial sums of series sometimes have simpler closed form expressions, for instance an [[arithmetic series]] has partial sums
<math display=block>
s_n = \sum_{k=0}^{n} \left(a + kd\right)
= a + (a + d) + (a + 2d) + \cdots + (a + nd)
= (n+1)\bigl(a + \tfrac12 n d\bigr),
</math>
and a [[geometric series]] has partial sums<ref name=":45">{{harvnb|Spivak|2008|pp=473–478}}</ref><ref name=":24">{{harvnb|Apostol|1967|pp=388–390, 399-401}}</ref><ref name=":16">{{harvnb|Rudin|1976|p=61}}</ref>
<math display=block>s_n = \sum_{k=0}^{n} ar^k = a + ar + ar^2 + \cdots + ar^n  = a\frac{1 - r^{n+1}}{1 - r}</math>
if {{tmath|r \neq 1}} or simply {{tmath|1= s_n = a(n+1)}} if {{tmath|1= r = 1}}.

===Sum of a series===
[[File:Geometric sequences.svg|thumb|right|Illustration of 3 [[geometric series]] with partial sums from 1 to 6 terms. The dashed line represents the limit.]]
Strictly speaking, a series is said to [[Convergent series|''converge'']], to be ''convergent'', or to be ''summable'' when the sequence of its partial sums has a  [[Limit of a sequence|limit]]. When the limit of the sequence of partial sums  does not exist, the series [[Divergent series|''diverges'']] or is ''divergent''.<ref name=":43">{{harvnb|Spivak|2008|p=453}}</ref> When the limit of the partial sums exists, it is called the ''sum of the series'' or ''value of the series'':<ref name=":4" /><ref name=":2" /><ref name=":3" /><ref name=":15" />
<math display=block>\sum_{k = 0}^\infty a_k = \lim_{n\to\infty} \sum_{k=0}^n a_k = \lim_{n\to\infty} s_n.</math>
A series with only a finite number of nonzero terms is always convergent. Such series are useful for considering finite sums without taking care of the numbers of terms.<ref>{{Cite journal |last=Knuth |first=Donald E. |author-link=Donald Knuth |year=1992 |title=Two Notes on Notation |journal=American Mathematical Monthly |volume=99 |issue=5 |pages=403–422 |doi=10.2307/2325085 |jstor=2325085 }}</ref> When the sum exists, the difference between the sum of a series and its <math>n</math>th partial sum, <math display=inline>s - s_n = \sum_{k=n+1}^\infty a_k,</math> is known as the <math>n</math>th ''[[truncation error]]'' of the infinite series.<ref name="Atkinson">{{Cite book |last=Atkinson |first=Kendall E. |title=An Introduction to Numerical Analysis |year=1989 |publisher=Wiley |isbn=978-0-471-62489-9 |edition=2nd |location=New York |page=20 |language=English |oclc=803318878}}</ref><ref name="Stoer">{{cite book |last1=Stoer |first1=Josef |title=Introduction to Numerical Analysis |year=2002 |edition=3rd |place=Princeton, N.J. |publisher=Recording for the Blind & Dyslexic |language=English |oclc=50556273 |last2=Bulirsch |first2=Roland}}</ref>

An example of a convergent series is the geometric series
<math display=block> 1 + \frac{1}{2}+ \frac{1}{4}+ \frac{1}{8} + \cdots + \frac{1}{2^k} + \cdots.</math>

It can be shown by algebraic computation that each partial sum <math>s_n</math> is 
<math display=block>\sum_{k=0}^n \frac 1{2^k} = 2-\frac 1{2^n}.</math> As one has 
<math display=block>\lim_{n \to \infty} \left(2-\frac 1{2^n}\right) =2,</math>
the series is convergent and converges to {{tmath|2}} with truncation errors <math display=inline> 1 / 2^n </math>.<ref name=":45" /><ref name=":24" /><ref name=":16" />

By contrast, the geometric series
<math display=block>\sum_{k = 0}^\infty 2^k</math>
is divergent in the [[real number]]s.<ref name=":45" /><ref name=":24" /><ref name=":16" /> However, it is convergent in the [[extended real number line]], with <math>+\infty</math> as its limit and <math>+\infty</math> as its truncation error at every step.<ref>{{Cite web |last=Wilkins |first=David |year=2007 |title=Section 6: The Extended Real Number System |url=https://www.maths.tcd.ie/~dwilkins/Courses/221/Extended.pdf |access-date=2019-12-03 |website=maths.tcd.ie}}</ref>

When a series's sequence of partial sums is not easily calculated and evaluated for convergence directly, [[convergence tests]] can be used to prove that the series converges or diverges.

== Grouping and rearranging terms ==

===Grouping===
In ordinary [[Finite summation|finite summations]], terms of the summation can be grouped and ungrouped freely without changing the result of the summation as a consequence of the [[associativity]] of addition. <math>a_0 + a_1 + a_2 = {}</math><math>a_0 + (a_1 + a_2) = {}</math><math>(a_0 + a_1) + a_2.</math> Similarly, in a series, any finite groupings of terms of the series will not change the limit of the partial sums of the series and thus will not change the sum of the series. However, if an infinite number of groupings is performed in an infinite series, then the partial sums of the grouped series may have a different limit than the original series and different groupings may have different limits from one another; the sum of <math>a_0 + a_1 + a_2 + \cdots</math> may not equal the sum of <math>a_0 + (a_1 + a_2) + {}</math><math>(a_3 + a_4) + \cdots.</math> 

For example, [[Grandi's series]] {{tmath|1-1+1-1+ \cdots}} has a sequence of partial sums that alternates back and forth between {{tmath|1}} and {{tmath|0}} and does not converge. Grouping its elements in pairs creates the series <math>(1 - 1) + (1 - 1) + (1 - 1) + \cdots = {}</math><math>0 + 0 + 0 + \cdots,</math> which has partial sums equal to zero at every term and thus sums to zero. Grouping its elements in pairs starting after the first creates the series <math>1 + (- 1 + 1) + {}</math><math>(- 1 + 1) + \cdots = {}</math><math>1 + 0 + 0 + \cdots,</math> which has partial sums equal to one for every term and thus sums to one, a different result.

In general, grouping the terms of a series creates a new series with a sequence of partial sums that is a [[subsequence]] of the partial sums of the original series. This means that if the original series converges, so does the new series after grouping: all infinite subsequences of a convergent sequence also converge to the same limit. However, if the original series diverges, then the grouped series do not necessarily diverge, as in this example of Grandi's series above. However, divergence of a grouped series does imply the original series must be divergent, since it proves there is a subsequence of the partial sums of the original series which is not convergent, which would be impossible if it were convergent. This reasoning was applied in [[Harmonic series (mathematics)#Comparison test|Oresme's proof of the divergence of the harmonic series]],<ref name=":0">{{Cite journal |last1=Kifowit |first1=Steven J. |last2=Stamps |first2=Terra A. |year= 2006 |title=The harmonic series diverges again and again |url=https://stevekifowit.com/pubs/harmapa.pdf |journal=American Mathematical Association of Two-Year Colleges Review |volume=27 |issue=2 |pages=31–43}}</ref> and it is the basis for the general [[Cauchy condensation test]].<ref name=":14" /><ref name=":17">{{harvnb|Rudin|1976|p=61}}</ref>

=== Rearrangement ===
In ordinary finite summations, terms of the summation can be rearranged freely without changing the result of the summation as a consequence of the [[commutativity]] of addition. <math>a_0 + a_1 + a_2 = {}</math><math>a_0 + a_2 + a_1 = {}</math><math>a_2 + a_1 + a_0.</math> Similarly, in a series, any finite rearrangements of terms of a series does not change the limit of the partial sums of the series and thus does not change the sum of the series: for any finite rearrangement, there will be some term after which the rearrangement did not affect any further terms: any effects of rearrangement can be isolated to the finite summation up to that term, and finite summations do not change under rearrangement.

However, as for grouping, an infinitary rearrangement of terms of a series can sometimes lead to a change in the limit of the partial sums of the series. Series with sequences of partial sums that converge to a value but whose terms could be rearranged to a form a series with partial sums that converge to some other value are called [[conditionally convergent]] series. Those that converge to the same value regardless of rearrangement are called [[unconditionally convergent]] series.

For series of real numbers and complex numbers, a series <math>a_0 + a_1 + a_2 + \cdots</math> is unconditionally convergent [[if and only if]] the series summing the [[Absolute value|absolute values]] of its terms, <math>|a_0| + |a_1| + |a_2| + \cdots, </math> is also convergent, a property called [[absolute convergence]]. Otherwise, any series of real numbers or complex numbers that converges but does not converge absolutely is conditionally convergent. Any conditionally convergent sum of real numbers can be rearranged to yield any other real number as a limit, or to diverge. These claims are the content of the [[Riemann series theorem]].<ref name=":46">{{harvnb|Spivak|2008|pp=483–486}}</ref><ref name=":25">{{harvnb|Apostol|1967|pp=412–414}}</ref><ref>{{harvnb|Rudin|1976|p=76}}</ref>

A historically important example of conditional convergence is the [[alternating harmonic series]],

<math display=block>\sum\limits_{n=1}^\infty {(-1)^{n+1}  \over n} = 1 - {1 \over 2} + {1 \over 3} - {1 \over 4} + {1 \over 5} - \cdots,</math>
which has a sum of the [[natural logarithm of 2]], while the sum of the absolute values of the terms is the [[Harmonic series (mathematics)|harmonic series]],
<math display=block>\sum\limits_{n=1}^\infty {1 \over n} = 1 + {1 \over 2} + {1 \over 3} + {1 \over 4} + {1 \over 5} + \cdots,</math>
which diverges per the divergence of the harmonic series,<ref name=":0" /> so the alternating harmonic series is conditionally convergent. For instance, rearranging the terms of the alternating harmonic series so that each positive term of the original series is followed by two negative terms of the original series rather than just one yields<ref name=":4222">{{harvnb|Spivak|2008|p=482}}</ref> 
<math display=block>\begin{align}
&1 - \frac12 - \frac14 + \frac13 - \frac16 - \frac18
  + \frac15 - \frac1{10} - \frac1{12} + \cdots \\[3mu]
&\quad = \left(1 - \frac12\right) - \frac14
  + \left(\frac13 - \frac16\right) - \frac18
  + \left(\frac15 - \frac1{10}\right) - \frac1{12} + \cdots \\[3mu]
&\quad = \frac12 - \frac14 + \frac16 - \frac18
  + \frac1{10} - \frac1{12} + \cdots \\[3mu]
&\quad = \frac12 \left(1 - \frac12 + \frac13 - \frac14
  + \frac15 - \frac16 + \cdots \right) ,
\end{align}</math>
which is <math>\tfrac12</math> times the original series, so it would have a sum of half of the natural logarithm of 2. By the Riemann series theorem, rearrangements of the alternating harmonic series to yield any other real number are also possible.

== Operations ==

=== Series addition ===
The addition of two series <math display=inline>a_0 + a_1 + a_2 + \cdots </math> and <math display=inline>b_0 + b_1 + b_2 + \cdots </math> is given by the termwise sum<ref name=":422" /><ref name=":242">{{harvnb|Apostol|1967|pp=385–386}}</ref><ref name=":9">{{Cite book |last1=Saff |first1=E. B. |title=Fundamentals of Complex Analysis |last2=Snider |first2=Arthur D. |publisher=Pearson Education |year=2003 |isbn=0-13-907874-6 |edition=3rd |pages=247–249}}</ref><ref>{{harvnb|Rudin|1976|p=72}}</ref> <math display=inline>(a_0 + b_0) + (a_1 + b_1) + (a_2 + b_2) + \cdots \,</math>, or, in summation notation,
<math display=block>\sum_{k=0}^{\infty} a_k + \sum_{k=0}^{\infty} b_k  = \sum_{k=0}^{\infty} a_k + b_k.  </math> 

Using the symbols <math>s_{a, n}  </math> and <math>s_{b, n}  </math> for the partial sums of the added series and <math>s_{a + b, n}  </math> for the partial sums of the resulting series, this definition implies the partial sums of the resulting series follow <math>s_{a + b, n} = s_{a, n} + s_{b, n}.</math> Then the sum of the resulting series, i.e., the limit of the sequence of partial sums of the resulting series, satisfies
<math display=block>\lim_{n \rightarrow \infty} s_{a + b, n} = \lim_{n \rightarrow \infty} (s_{a, n} + s_{b, n}) = \lim_{n \rightarrow \infty} s_{a, n} + \lim_{n \rightarrow \infty} s_{b , n},</math>
when the limits exist. Therefore, first, the series resulting from addition is summable if the series added were summable, and, second, the sum of the resulting series is the addition of the sums of the added series. The addition of two divergent series may yield a convergent series: for instance, the addition of a divergent series with a series of its terms times <math>-1</math> will yield a series of all zeros that converges to zero. However, for any two series where one converges and the other diverges, the result of their addition diverges.<ref name=":242" />

For series of real numbers or complex numbers, series addition is [[Associative property|associative]], [[Commutative property|commutative]], and [[invertible]]. Therefore series addition gives the sets of convergent series of real numbers or complex numbers the structure of an [[abelian group]] and also gives the sets of all series of real numbers or complex numbers (regardless of convergence properties) the structure of an abelian group.

=== Scalar multiplication ===
The product of a series <math display=inline>a_0 + a_1 + a_2 + \cdots </math> with a constant number <math>c</math>, called a [[Scalar (mathematics)|scalar]] in this context, is given by the termwise product<ref name=":242" /> <math display=inline>ca_0 + ca_1 + ca_2 + \cdots </math>, or, in summation notation,

<math display=block>c\sum_{k=0}^{\infty} a_k = \sum_{k=0}^{\infty} ca_k.  </math>

Using the symbols <math>s_{a, n}  </math> for the partial sums of the original series and <math>s_{ca, n}  </math> for the partial sums of the series after multiplication by <math>c</math>, this definition implies that <math>s_{ca, n} = c s_{a, n}  </math> for all <math>n,  </math> and therefore also <math display=inline>\lim_{n \rightarrow \infty} s_{ca, n} = c \lim_{n \rightarrow \infty} s_{a, n},  </math>when the limits exist. Therefore if a series is summable, any nonzero scalar multiple of the series is also summable and vice versa: if a series is divergent, then any nonzero scalar multiple of it is also divergent.

Scalar multiplication of real numbers and complex numbers is associative, commutative, invertible, and it [[Distributive property|distributes over]] series addition.  

In summary, series addition and scalar multiplication gives the set of convergent series and the set of series of real numbers the structure of a [[real vector space]]. Similarly, one gets [[complex vector space]]s for series and convergent series of complex numbers. All these vector spaces are infinite dimensional.

=== Series multiplication ===
The multiplication of two series <math>a_0 + a_1 + a_2 + \cdots </math> and <math>b_0 + b_1 + b_2 + \cdots </math> to generate a third series <math>c_0 + c_1 + c_2 + \cdots </math>, called the Cauchy product,<ref name=":7" /><ref name=":422" /><ref name=":8" /><ref name=":9" /><ref>{{harvnb|Rudin|1976|p=73}}</ref> can be written in summation notation
<math display=block>
\biggl( \sum_{k=0}^{\infty} a_k \biggr) \cdot  \biggl( \sum_{k=0}^{\infty} b_k \biggr)
= \sum_{k=0}^{\infty} c_k = \sum_{k=0}^{\infty} \sum_{j=0}^{k} a_{j} b_{k-j},
</math>
with each <math display=inline>c_k = \sum_{j=0}^{k} a_{j} b_{k-j} = {}\!</math><math>\!a_0 b_k + a_1 b_{k-1} + \cdots + a_{k-1} b_1 + a_k b_0.</math> Here, the convergence of the partial sums of the series <math>c_0 + c_1 + c_2 + \cdots </math> is not as simple to establish as for addition. However, if both series <math>a_0 + a_1 + a_2 + \cdots </math> and <math>b_0 + b_1 + b_2 + \cdots </math> are [[absolutely convergent]] series, then the series resulting from multiplying them also converges absolutely with a sum equal to the product of the two sums of the multiplied series,<ref name=":422" /><ref name=":9" /><ref>{{harvnb|Rudin|1976|p=74}}</ref>
<math display=block>\lim_{n \rightarrow \infty} s_{c, n} = \left(\, \lim_{n \rightarrow \infty} s_{a, n} \right) \cdot \left(\, \lim_{n \rightarrow \infty} s_{b , n} \right).</math>

Series multiplication of absolutely convergent series of real numbers and complex numbers is associative, commutative, and distributes over series addition. Together with series addition, series multiplication gives the sets of absolutely convergent series of real numbers or complex numbers the structure of a [[Commutative ring|commutative]] [[Ring (mathematics)|ring]], and together with scalar multiplication as well, the structure of a [[Commutative algebra (structure)|commutative algebra]]; these operations also give the sets of all series of real numbers or complex numbers the structure of an [[associative algebra]].

==Examples of numerical series==
{{For|other examples|List of mathematical series|Sums of reciprocals#Infinitely many terms}}
* A ''[[geometric series]]''<ref name=":45" /><ref name=":24" /> is one where each successive term is produced by multiplying the previous term by a constant number (called the common ratio in this context). For example: <math display=block>
1 + {1 \over 2} + {1 \over 4} + {1 \over 8} + {1 \over 16} + \cdots=\sum_{n=0}^\infty{1 \over 2^n} = 2.
</math> In general, a geometric series with initial term <math> a</math> and common ratio <math> r</math>, <math display=inline>\sum_{n=0}^\infty a r^n,</math> converges if and only if <math display=inline>|r| < 1</math>, in which case it converges to <math display=inline> {a \over 1 - r}</math>.
* The ''[[harmonic series (mathematics)|harmonic series]]'' is the series<ref name=":243">{{harvnb|Apostol|1967|p=384}}</ref> <math display=block>1 + {1 \over 2} + {1 \over 3} + {1 \over 4} + {1 \over 5} + \cdots = \sum_{n=1}^\infty {1 \over n}.</math> The harmonic series is [[harmonic series (mathematics)#Divergence|divergent]].
* An ''[[alternating series]]'' is a series where terms alternate signs.<ref name=":2434">{{harvnb|Apostol|1967|pp=403–404}}</ref> Examples: <math display=block>
1 - {1 \over 2} + {1 \over 3} - {1 \over 4} + {1 \over 5} - \cdots
= \sum_{n=1}^\infty {\left(-1\right)^{n-1} \over n} = \ln(2),
</math> the [[alternating harmonic series]], and <math display=block>
-1+\frac{1}{3} - \frac{1}{5} + \frac{1}{7} - \frac{1}{9} + \cdots
= \sum_{n=1}^\infty \frac{\left(-1\right)^n}{2n-1} = -\frac{\pi}{4},
</math> the [[Leibniz formula for π|Leibniz formula for <math>\pi.</math>]]
* A [[telescoping series]]<ref name=":2432">{{harvnb|Apostol|1967|p=386}}</ref> <math display=block>
\sum_{n=1}^\infty \left(b_n-b_{n+1}\right)
</math> converges if the [[sequence]] {{tmath|b_n}} converges to a limit {{tmath|L}} as {{tmath|n}} goes to infinity. The value of the series is then {{tmath|b_1 - L}}.<ref name=":10">{{harvnb|Apostol|1967|p=387}}</ref>
* An ''[[arithmetico-geometric series]]'' is a series that has terms which are each the product of an element of an [[arithmetic progression]] with the corresponding element of a [[geometric progression]]. Example: <math display=block>3 + {5 \over 2} + {7 \over 4} + {9 \over 8} + {11 \over 16} + \cdots=\sum_{n=0}^\infty{(3+2n) \over 2^n}.</math>
* The [[Dirichlet series]] <math display=block>
\sum_{n=1}^\infty\frac{1}{n^p}
</math> converges for {{tmath|p>1}} and diverges for {{tmath|p \leq 1}}, which can be shown with the [[integral test for convergence]] described below in [[Series (mathematics)#Convergence tests|convergence tests]]. As a function of {{tmath|p}}, the sum of this series is [[Riemann zeta function|Riemann's zeta function]].<ref name=":2433">{{harvnb|Apostol|1967|p=396}}</ref>
* [[Hypergeometric series]]: <math display="block">
_pF_q \left[ \begin{matrix}a_1, a_2, \dotsc, a_p \\ b_1, b_2, \dotsc, b_q \end{matrix}; z \right]
:= \sum_{n=0}^{\infty} \frac{\prod_{r=1}^{p} (a_r)_n}{\prod_{s=1}^{q} (b_s)_n} \frac{z^n}{n!}
</math> and their generalizations (such as [[basic hypergeometric series]] and [[elliptic hypergeometric series]]) frequently appear in [[integrable systems]] and [[mathematical physics]].<ref>Gasper, G., Rahman, M. (2004). Basic hypergeometric series. [[Cambridge University Press]].</ref>
* There are some elementary series whose convergence is not yet known/proven. For example, it is unknown whether the Flint Hills series, <math display=block>
\sum_{n=1}^\infty \frac{1}{n^{3}\sin^{2} n},
</math> converges or not. The convergence depends on how well <math>\pi</math> can be approximated with [[rational numbers]] (which is unknown as of yet). More specifically, the values of {{tmath|n}} with large numerical contributions to the sum are the numerators of the continued fraction convergents of <math>\pi</math>, a sequence beginning with 1, 3, 22, 333, 355, 103993, ... {{OEIS|A046947}}. These are integers {{tmath|n}} that are close to <math>m\pi</math> for some integer {{tmath|m}}, so that <math>\sin n</math> is close to <math>\sin m\pi = 0</math> and its reciprocal is large.

===Pi===
{{Main|Basel problem|Leibniz formula for π}}

<math display=block> \sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots = \frac{\pi^2}{6}</math>

<math display="block"> 4\sum_{n=1}^\infty \frac{(-1)^{n+1}}{2n-1} = \frac{4}{1} - \frac{4}{3} + \frac{4}{5} - \frac{4}{7} + \frac{4}{9} - \frac{4}{11} + \frac{4}{13} - \cdots = \pi</math>

===Natural logarithm of 2===
{{Main|Natural logarithm of 2#Series representations}}

<math display=block>\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} = \ln 2</math>

<math display=block> \sum_{n=1}^\infty \frac{1}{2^{n}n} = \ln 2</math>

===Natural logarithm base {{mvar|e}} ===
{{Main|e (mathematical constant)}}

<math display=block>\sum_{n = 0}^\infty \frac{(-1)^n}{n!} =  1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\cdots = \frac{1}{e}</math>

<math display=block> \sum_{n = 0}^\infty \frac{1}{n!} = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \cdots = e </math>

==Convergence testing==
{{Main|Convergence tests}}

One of the simplest tests for convergence of a series, applicable to all series, is the ''vanishing condition'' or ''[[Nth-term test|{{tmath|n}}th-term test]]'': If <math display=inline>\lim_{n \to \infty} a_n \neq 0</math>, then the series diverges; if <math display=inline>\lim_{n \to \infty} a_n = 0</math>, then the test is inconclusive.<ref>{{harvnb|Spivak|2008|p=473}}</ref><ref name=":18">{{harvnb|Rudin|1976|p=60}}</ref>

===Absolute convergence tests===
{{Main|Absolute convergence}}

When every term of a series is a non-negative real number, for instance when the terms are the [[Absolute value|absolute values]] of another series of real numbers or complex numbers, the sequence of partial sums is non-decreasing. Therefore a series with non-negative terms converges if and only if the sequence of partial sums is bounded, and so finding a bound for a series or for the absolute values of its terms is an effective way to prove convergence or absolute convergence of a series.<ref>{{harvnb|Apostol|1967|pp=381,394-395}}</ref><ref>{{harvnb|Spivak|2008|pp=457,473-474}}</ref><ref name=":18" /><ref>{{harvnb|Rudin|1976|pp=71-72}}</ref> 

For example, the series <math display=inline>1 + \frac14 + \frac19 + \cdots + \frac1{n^2} + \cdots\,</math>is convergent and absolutely convergent because <math display=inline>\frac1{n^2} \le \frac1{n-1} - \frac1n</math> for all <math>n \geq 2</math> and a [[telescoping sum]] argument implies that the partial sums of the series of those non-negative bounding terms are themselves bounded above by 2.<ref name=":10" /> The exact value of this series is <math display=inline>\frac16\pi^2</math>; see [[Basel problem]].

This type of bounding strategy is the basis for general series comparison tests. First is the general ''[[direct comparison test]]'':<ref>{{harvnb|Apostol|1967|pp=395–396}}</ref><ref>{{harvnb|Spivak|2008|pp=474–475}}</ref><ref name=":18" /> For any series <math display=inline>\sum a_n</math>, If <math display=inline>\sum b_n</math> is an [[absolute convergence|absolutely convergent]] series such that <math>\left\vert a_n \right\vert \leq C \left\vert b_n \right\vert</math> for some positive real number <math>C</math> and for sufficiently large <math>n</math>, then <math display=inline>\sum a_n</math> converges absolutely as well. If <math display=inline>\sum \left\vert b_n \right\vert</math> diverges, and <math>\left\vert a_n \right\vert \geq \left\vert b_n \right\vert</math>  for all sufficiently large <math>n</math>, then <math display=inline>\sum a_n</math> also fails to converge absolutely, although it could still be conditionally convergent, for example, if the <math>a_n</math> alternate in sign. Second is the general ''[[limit comparison test]]'':<ref>{{harvnb|Apostol|1967|p=396}}</ref><ref>{{harvnb|Spivak|2008|p=475–476}}</ref> If <math display=inline>\sum b_n</math> is an absolutely convergent series such that <math>\left\vert \tfrac{a_{n+1}}{a_{n}} \right\vert \leq \left\vert \tfrac{b_{n+1}}{b_{n}} \right\vert</math> for sufficiently large <math>n</math>, then <math display=inline>\sum a_n</math> converges absolutely as well. If <math display=inline>\sum \left| b_n \right|</math> diverges, and <math>\left\vert \tfrac{a_{n+1}}{a_{n}} \right\vert \geq \left\vert \tfrac{b_{n+1}}{b_{n}} \right\vert</math> for all sufficiently large <math>n</math>, then <math display=inline>\sum a_n</math> also fails to converge absolutely, though it could still be conditionally convergent if the <math>a_n</math> vary in sign.

Using comparisons to [[geometric series]] specifically,<ref name=":45" /><ref name=":24" /> those two general comparison tests imply two further common and generally useful tests for convergence of series with non-negative terms or for absolute convergence of series with general terms. First is the ''[[ratio test]]'':<ref name=":11">{{harvnb|Apostol|1967|pp=399–401}}</ref><ref>{{harvnb|Spivak|2008|pp=476–478}}</ref><ref>{{harvnb|Rudin|1976|p=66}}</ref> if there exists a constant <math>C < 1</math> such that <math>\left\vert \tfrac{a_{n+1}}{a_{n}} \right\vert < C</math> for all sufficiently large&nbsp;<math>n</math>, then <math display=inline>\sum a_{n}</math> converges absolutely. When the ratio is less than <math>1</math>, but not less than a constant less than <math>1</math>, convergence is possible but this test does not establish it. Second is the ''[[root test]]'':<ref name=":11" /><ref>{{harvnb|Spivak|2008|p=493}}</ref><ref>{{harvnb|Rudin|1976|p=65}}</ref> if there exists a constant <math>C < 1</math> such that <math>\textstyle \left\vert a_{n} \right\vert^{1/n} \leq C</math> for all sufficiently large&nbsp;<math>n</math>, then <math display=inline>\sum a_{n}</math> converges absolutely.

Alternatively, using comparisons to series representations of [[Integral|integrals]] specifically, one derives the [[Integral test for convergence|''integral test'']]:<ref>{{harvnb|Apostol|1967|pp=397–398}}</ref><ref>{{harvnb|Spivak|2008|pp=478–479}}</ref> if <math>f(x)</math> is a positive [[monotone decreasing]] function defined on the [[interval (mathematics)|interval]] <math>[1,\infty)</math> <!--DO NOT "FIX" THE "TYPO" IN THE FOREGOING.  IT IS INTENDED TO SAY [...) WITH A SQUARE BRACKET ON THE LEFT AND A ROUND BRACKET ON THE RIGHT. --> then for a series with terms <math>a_n = f(n)</math> for all&nbsp;<math>n</math>, <math display=inline>\sum a_{n}</math> converges if and only if the [[integral]] <math display=inline>\int_{1}^{\infty} f(x) \, dx</math> is finite. Using comparisons to flattened-out versions of a series leads to [[Cauchy's condensation test]]:<ref name=":14">{{harvnb|Spivak|2008|p=496}}</ref><ref name=":17" /> if the sequence of terms <math>a_{n}</math> is non-negative and non-increasing, then the two series <math display=inline>\sum a_{n}</math> and <math display=inline>\sum 2^{k} a_{(2^{k})}</math> are either both convergent or both divergent.

===Conditional convergence tests===
{{Main|Conditional convergence}}
A series of real or complex numbers is said to be ''conditionally convergent'' (or ''semi-convergent'') if it is convergent but not absolutely convergent. Conditional convergence is tested for differently than absolute convergence.

One important example of a test for conditional convergence is the ''[[alternating series test]]'' or ''Leibniz test'':<ref>{{harvnb|Apostol|1967|pp=403–404}}</ref><ref>{{harvnb|Spivak|2008|p=481}}</ref><ref>{{harvnb|Rudin|1976|p=71}}</ref> A series of the form <math display=inline>\sum (-1)^{n} a_{n}</math> with all <math>a_{n} > 0</math> is called ''alternating''. Such a series converges if the non-negative [[sequence]] <math>a_{n}</math> is [[monotone decreasing]] and converges to&nbsp;<math>0</math>. The converse is in general not true. A famous example of an application of this test is the [[alternating harmonic series]]
<math display=block>\sum\limits_{n=1}^\infty {(-1)^{n+1}  \over n} = 1 - {1 \over 2} + {1 \over 3} - {1 \over 4} + {1 \over 5} - \cdots,</math>
which is convergent per the alternating series test (and its sum is equal to&nbsp;<math>\ln 2</math>), though the series formed by taking the absolute value of each term is the ordinary [[Harmonic series (mathematics)|harmonic series]], which is divergent.<ref name=":23">{{harvnb|Apostol|1967|pp=413–414}}</ref><ref name=":44">{{harvnb|Spivak|2008|pp=482–483}}</ref>

The alternating series test can be viewed as a special case of the more general ''[[Dirichlet's test]]'':<ref name=":13" /><ref>{{harvnb|Spivak|2008|p=495}}</ref><ref>{{harvnb|Rudin|1976|p=70}}</ref> if <math>(a_{n})</math> is a sequence of terms of decreasing nonnegative real numbers that converges to zero, and <math>(\lambda_n)</math> is a sequence of terms with bounded partial sums, then the series <math display=inline>\sum \lambda_n a_n </math> converges. Taking <math>\lambda_n = (-1)^n</math> recovers the alternating series test.

''[[Abel's test]]'' is another important technique for handling semi-convergent series.<ref name=":13">{{harvnb|Apostol|1967|pp=407–409}}</ref><ref name=":14" /> If a series has the form <math display=inline>\sum a_n = \sum \lambda_n b_n</math> where the partial sums of the series with terms <math>b_n</math>, <math>s_{b,n} = b_{0} + \cdots + b_{n}</math> are bounded, <math>\lambda_{n}</math> has [[bounded variation]], and <math>\lim \lambda_{n} b_{n}</math> exists: if <math display=inline>\sup_n |s_{b,n}| < \infty,</math> <math display=inline>\sum \left|\lambda_{n+1} - \lambda_n\right| < \infty,</math> and <math>\lambda_n s_{b,n}</math>converges, then the series <math display=inline>\sum a_{n}</math> is convergent.

Other specialized convergence tests for specific types of series include the [[Dini test]]<ref>{{harvnb|Spivak|2008|p=524}}</ref> for [[Fourier series]].

===Evaluation of truncation errors===
The evaluation of truncation errors of series is important in [[numerical analysis]] (especially [[validated numerics]] and [[computer-assisted proof]]). It can be used to prove convergence and to analyze [[Rate of convergence|rates of convergence]].

====Alternating series====
{{Main|Alternating series}}
When conditions of the [[alternating series test]] are satisfied by <math display=inline>S:=\sum_{m=0}^\infty(-1)^m u_m</math>, there is an exact error evaluation.<ref>[https://www.ck12.org/book/CK-12-Calculus-Concepts/section/9.9/ Positive and Negative Terms: Alternating Series]</ref> Set <math>s_n</math> to be the partial sum <math display=inline>s_n:=\sum_{m=0}^n(-1)^m u_m</math> of the given alternating series <math>S</math>. Then the next inequality holds:
<math display=block>|S-s_n|\leq u_{n+1}.</math>

====Hypergeometric series====
{{Main|Hypergeometric series}}
By using the [[ratio]], we can obtain the evaluation of the error term when the [[hypergeometric series]] is truncated.<ref>Johansson, F. (2016). Computing hypergeometric functions rigorously. arXiv preprint arXiv:1606.06977.</ref>

====Matrix exponential====
{{Main|Matrix exponential}}
For the [[matrix exponential]]:

<math display=block>\exp(X) := \sum_{k=0}^\infty\frac{1}{k!}X^k,\quad X\in\mathbb{C}^{n\times n},</math>

the following error evaluation holds (scaling and squaring method):<ref>Higham, N. J. (2008). Functions of matrices: theory and computation. [[Society for Industrial and Applied Mathematics]].</ref><ref>Higham, N. J. (2009). The scaling and squaring method for the matrix exponential revisited. SIAM review, 51(4), 747-764.</ref><ref>[http://www.maths.manchester.ac.uk/~higham/talks/exp10.pdf How and How Not to Compute the Exponential of a Matrix]</ref>

<math display=block>T_{r,s}(X) := \biggl(\sum_{j=0}^r\frac{1}{j!}(X/s)^j\biggr)^s,\quad \bigl\|\exp(X)-T_{r,s}(X)\bigr\|\leq\frac{\|X\|^{r+1}}{s^r(r+1)!}\exp(\|X\|).</math>

== Sums of divergent series ==
{{Main|Divergent series}}
Under many circumstances, it is desirable to assign generalized sums to series which fail to converge in the strict sense that their sequences of partial sums do not converge. A ''[[summation method]]'' is any method for assigning sums to divergent series in a way that systematically extends the classical notion of the sum of a series. Summation methods include [[Cesàro summation]], [[Cesàro summation#(C, α) summation|generalized Cesàro {{tmath|(C, \alpha)}} summation]], [[Abel summation]], and [[Borel summation]], in order of applicability to increasingly divergent series. These methods are all based on [[Sequence transformation|sequence transformations]] of the original series of terms or of its sequence of partial sums.
A variety of general results concerning possible summability methods are known. The [[Silverman–Toeplitz theorem]] characterizes ''matrix summation methods'', which are methods for summing a divergent series by applying an infinite matrix to the vector of coefficients. The most general methods for summing a divergent series are [[non-constructive]] and concern [[Banach limit]]s.

==Series of functions==
{{Main|Function series}}
A series of real- or complex-valued functions

<math display=block>\sum_{n=0}^\infty f_n(x)</math>

is [[Pointwise convergence|pointwise convergent]] to a limit {{tmath|f(x)}} on a set {{tmath|E}} if the series converges for each {{tmath|x}} in {{tmath|E}} as a series of real or complex numbers. Equivalently, the partial sums

<math display=block>s_N(x) = \sum_{n=0}^N f_n(x)</math>

converge to {{tmath|f(x)}} as {{tmath|N}} goes to infinity for each {{tmath|x}} in {{tmath|E}}.

A stronger notion of convergence of a series of functions is [[uniform convergence]]. A series converges uniformly in a set <math>E</math> if it converges pointwise to the function {{tmath|f(x)}} at every point of <math>E</math> and the supremum of these pointwise errors in approximating the limit by the {{tmath|N}}th partial sum,

<math display=block>\sup_{x \in E} \bigl|s_N(x) - f(x)\bigr|</math>

converges to zero with increasing {{tmath|N}}, {{em|independently}} of {{tmath|x}}.

Uniform convergence is desirable for a series because many properties of the terms of the series are then retained by the limit.  For example, if a series of continuous functions converges uniformly, then the limit function is also continuous.  Similarly, if the {{tmath|f_n}} are [[integral|integrable]] on a closed and bounded interval {{tmath|I}} and converge uniformly, then the series is also integrable on {{tmath|I}} and can be integrated term by term. Tests for uniform convergence include [[Weierstrass M-test|Weierstrass' M-test]], [[Abel's uniform convergence test]], [[Dini's test]], and the [[Cauchy sequence|Cauchy criterion]].

More sophisticated types of convergence of a series of functions can also be defined. In [[measure theory]], for instance, a series of functions converges [[almost everywhere]] if it converges pointwise except on a set of [[null set|measure zero]]. Other [[modes of convergence]] depend on a different [[metric space]] structure on the [[Function space|space of functions]] under consideration. For instance, a series of functions '''converges in mean''' to a limit function {{tmath|f}} on a set {{tmath|E}} if

<math display=block>\lim_{N \rightarrow \infty} \int_E \bigl|s_N(x)-f(x)\bigr|^2\,dx = 0.</math>

===Power series===
:{{Main|Power series}}

A '''power series''' is a series of the form

<math display=block>\sum_{n=0}^\infty a_n(x-c)^n.</math>

The [[Taylor series]] at a point {{tmath|c}} of a function is a power series that, in many cases, converges to the function in a neighborhood of {{tmath|c}}. For example, the series

<math display=block>\sum_{n=0}^{\infty} \frac{x^n}{n!}</math>

is the Taylor series of <math>e^x</math> at the origin and converges to it for every {{tmath|x}}.

Unless it converges only at {{tmath|1= x = c}}, such a series converges on a certain open disc of convergence centered at the point {{tmath|c}} in the complex plane, and may also converge at some of the points of the boundary of the disc.  The radius of this disc is known as the [[radius of convergence]], and can in principle be determined from the asymptotics of the coefficients {{tmath|a_n}}.  The convergence is uniform on [[closed set|closed]] and [[bounded set|bounded]] (that is, [[compact set|compact]]) subsets of the interior of the disc of convergence: to wit, it is [[Compact convergence|uniformly convergent on compact sets]].

Historically, mathematicians such as [[Leonhard Euler]] operated liberally with infinite series, even if they were not convergent. When calculus was put on a sound and correct foundation in the nineteenth century, rigorous proofs of the convergence of series were always required.

=== Formal power series ===
{{main|Formal power series}}

While many uses of power series refer to their sums, it is also possible to treat power series as ''formal sums'', meaning that no addition operations are actually performed, and the symbol "+" is an abstract symbol of conjunction which is not necessarily interpreted as corresponding to addition. In this setting, the sequence of coefficients itself is of interest, rather than the convergence of the series. [[Formal power series]] are used in [[combinatorics]] to describe and study [[sequence]]s that are otherwise difficult to handle, for example, using the method of [[generating function]]s.  The [[Hilbert–Poincaré series]] is a formal power series used to study [[graded algebra]]s.

Even if the limit of the power series is not considered, if the terms support appropriate structure then it is possible to define operations such as [[addition]], [[multiplication]], [[derivative]], [[antiderivative]] for power series "formally", treating the symbol "+" as if it corresponded to addition. In the most common setting, the terms come from a [[commutative ring]], so that the formal power series can be added term-by-term and multiplied via the [[Cauchy product]]. In this case the algebra of formal power series is the [[total algebra]] of the [[monoid]] of [[natural numbers]] over the underlying term ring.<ref>{{citation|author=Nicolas Bourbaki|author-link=Nicolas Bourbaki|title=Algebra|publisher=Springer|year=1989}}: §III.2.11.</ref>  If the underlying term ring is a [[differential algebra]], then the algebra of formal power series is also a differential algebra, with differentiation performed term-by-term.

===Laurent series===
{{Main|Laurent series}}
[[Laurent series]] generalize power series by admitting terms into the series with negative as well as positive exponents.  A Laurent series is thus any series of the form

<math display=block>\sum_{n=-\infty}^\infty a_n x^n.</math>

If such a series converges, then in general it does so in an [[annulus (mathematics)|annulus]] rather than a disc, and possibly some boundary points.  The series converges uniformly on compact subsets of the interior of the annulus of convergence.

===Dirichlet series===
:{{Main|Dirichlet series}}

A [[Dirichlet series]] is one of the form

<math display=block>\sum_{n=1}^\infty {a_n \over n^s},</math>

where {{tmath|s}} is a [[complex number]].  For example, if all {{tmath|a_n}} are equal to {{tmath|1}}, then the sum of the Dirichlet series is the [[Riemann zeta function]]

<math display=block>\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}.</math>

Like the zeta function, Dirichlet series in general play an important role in [[analytic number theory]].  Generally a Dirichlet series converges if the real part of {{tmath|s}} is greater than a number called the abscissa of convergence.  In many cases, a function defined by a Dirichlet series is an [[analytic function]] that can be extended outside the domain of convergence of the series by [[analytic continuation]].  For example, the Dirichlet series for the zeta function converges absolutely when {{tmath|\operatorname{Re}(s)>1}}, but the zeta function can be extended to a [[holomorphic function]] defined on <math>\Complex\setminus\{1\}</math> with a simple [[pole (complex analysis)|pole]] at&nbsp;{{tmath|1}}.

This series can be directly generalized to [[general Dirichlet series]].

===Trigonometric series===
{{Main|Trigonometric series}}
A series of functions in which the terms are [[trigonometric function]]s is called a '''trigonometric series''':

<math display=block>A_0 + \sum_{n=1}^\infty \left(A_n\cos nx + B_n \sin nx\right).</math>

The most important example of a trigonometric series is the [[Fourier series]] of a function.

=== Asymptotic series ===
{{Main|Asymptotic expansion}}
[[Asymptotic series]], typically called [[asymptotic expansion]]s, are infinite series whose terms are functions of a sequence of different [[Big O notation|asymptotic orders]] and whose partial sums are approximations of some other function in an [[asymptotic limit]]. In general they do not converge, but they are still useful as sequences of approximations, each of which provides a value close to the desired answer for a finite number of terms. They are crucial tools in [[perturbation theory]] and in the [[analysis of algorithms]].

An asymptotic series cannot necessarily be made to produce an answer as exactly as desired away from the asymptotic limit, the way that an ordinary convergent series of functions can. In fact, a typical asymptotic series reaches its best practical approximation away from the asymptotic limit after a finite number of terms; if more terms are included, the series will produce less accurate approximations.

==History of the theory of infinite series==

===Development of infinite series===
Infinite series play an important role in modern analysis of [[Ancient Greece|Ancient Greek]] [[philosophy of motion]], particularly in [[Zeno's paradox|Zeno's paradoxes]].<ref name=":12">{{Citation |last=Huggett |first=Nick |title=Zeno's Paradoxes |year=2024 |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/archives/spr2024/entries/paradox-zeno/ |access-date=2024-03-25 |edition=Spring 2024 |publisher=Metaphysics Research Lab, Stanford University |editor2-last=Nodelman |editor2-first=Uri |encyclopedia=The Stanford Encyclopedia of Philosophy}}</ref> The paradox of [[Achilles and the tortoise]] demonstrates that continuous motion would require an [[actual infinity]] of temporal instants, which was arguably an [[absurdity]]: Achilles runs after a tortoise, but when he reaches the position of the tortoise at the beginning of the race, the tortoise has reached a second position; when he reaches this second position, the tortoise is at a third position, and so on. [[Zeno of Elea|Zeno]] is said to have argued that therefore Achilles could ''never'' reach the tortoise, and thus that continuous movement must be an illusion. Zeno divided the race into infinitely many sub-races, each requiring a finite amount of time, so that the total time for Achilles to catch the tortoise is given by a series. The resolution of the purely mathematical and imaginative side of the paradox is that, although the series has an infinite number of terms, it has a finite sum, which gives the time necessary for Achilles to catch up with the tortoise. However, in modern philosophy of motion the physical side of the problem remains open, with both philosophers and physicists doubting, like Zeno, that spatial motions are infinitely divisible: hypothetical reconciliations of [[quantum mechanics]] and [[general relativity]] in theories of [[quantum gravity]] often introduce [[Quantization (physics)|quantizations]] of [[spacetime]] at the [[Planck scale]].<ref>{{citation |last=Snyder |first=H. |title=Quantized space-time |journal=Physical Review |volume=67 |issue=1 |pages=38–41 |year=1947 |bibcode=1947PhRv...71...38S |doi=10.1103/PhysRev.71.38}}.</ref><ref>{{Cite web |date=2024-09-25 |title=The Unraveling of Space-Time |url=https://www.quantamagazine.org/the-unraveling-of-space-time-20240925/ |access-date=2024-10-11 |website=Quanta Magazine |language=en}}</ref>

[[Greek mathematics|Greek]] mathematician [[Archimedes]] produced the first known summation of an infinite series with a
method that is still used in the area of calculus today. He used the [[method of exhaustion]] to calculate the [[area]] under the arc of a [[parabola]] with the summation of an infinite series,<ref name=":6" /> and gave a remarkably accurate approximation of [[Pi|π]].<ref>{{cite web | title = A history of calculus |author1=O'Connor, J.J.  |author2=Robertson, E.F.  |name-list-style=amp | publisher = [[University of St Andrews]]| url = http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/The_rise_of_calculus.html |year=1996 |access-date= 2007-08-07}}</ref><ref>{{cite journal |title=Archimedes and Pi-Revisited. |last=Bidwell |first=James K. |date=30 November 1993 |journal=School Science and Mathematics |volume=94 |issue=3 |pages=127–129 |doi=10.1111/j.1949-8594.1994.tb15638.x }}</ref>

In the 14th century, French mathematician [[Nicole Oresme]] developed the first proof of the [[divergent series|divergence]] of the [[harmonic series (mathematics)|harmonic series]].<ref>{{cite book|author-link=Nicole Oresme|first=Nicole|last=Oresme|date=c. 1360|title=Quaestiones super Geometriam Euclidis|trans-title=Questions concerning Euclid's Geometry}}</ref> His work, along with the contemporaneous work of [[Richard Swineshead]] on a different series, marked the first appearance of infinite series other than the [[geometric series]] in mathematics.<ref>{{cite journal |last=Stillwell |first=John |date=2010 |title=Mathematics and Its History |journal=Undergraduate Texts in Mathematics |doi=10.1007/978-1-4419-6053-5 |issn=0172-6056}}</ref>

Mathematicians from the [[Kerala school of astronomy and mathematics|Kerala school]] in [[medieval India]] were studying infinite series {{circa|1350 CE}}. One of their most important works—series expansion for trigonometric functions—were described in [[Sanskrit]] verse in a book by Neelakanta called ''[[Tantrasangraha]]'' (around 1500), and again in a commentary on this work, called ''Tantrasangraha-vakhya'', of unknown authorship. The theorems were stated without proof, but proofs for the series for sine, cosine, and inverse tangent were provided a century later in the work ''[[Yuktibhasa]]'' (<abbr>c.</abbr> 1530), written in [[Malayalam]], by Jyesthadeva, and also in a commentary on ''Tantrasangraha''.<ref>Stillwell, John (2004), ''Mathematics and its History'' (2 ed.), Berlin and New York: Springer, 568 pages,   {{ISBN|978-0-387-95336-6}}</ref><ref>Bressoud, David (2002), "Was Calculus Invented in India?", ''The College Mathematics Journal'', '''33''' (1): 2–13, {{doi|10.2307/1558972}}, {{JSTOR|1558972}} 

Quote: "There is no evidence that the Indian work on series was known beyond India, or even outside Kerala, until the nineteenth century. Gold and Pingree assert [4] that by the time these series were rediscovered in Europe, they had, for all practical purposes, been lost to India. The expansions of the sine, cosine, and arc tangent had been passed down through several generations of disciples, but they remained sterile observations for which no one could find much use."</ref><ref>[[Kim Plofker|Plofker, Kim]] (2001), "The "Error" in the Indian "Taylor Series Approximation" to the Sine", ''Historia Mathematica'', '''28''' (4): 283–295, {{doi|10.1006/hmat.2001.2331}}

p. 293 Quote: "It is not unusual to encounter in discussions of Indian mathematics such assertions as that "the concept of differentiation was understood [in India] from the time of Manjula (... in the 10th century)" [Joseph 1991, 300], or that "we may consider Madhava to have been the founder of mathematical analysis" (Joseph 1991, 293), or that Bhaskara II may claim to be "the precursor of Newton and Leibniz in the discovery of the principle of the differential calculus" (Bag 1979, 294). ... The points of resemblance, particularly between early European calculus and the Keralese work on power series, have even inspired suggestions of a possible transmission of mathematical ideas from the Malabar coast in or after the 15th century to the Latin scholarly world (e.g., in (Bag 1979, 285)). ... It should be borne in mind, however, that such an emphasis on the similarity of Sanskrit (or Malayalam) and Latin mathematics risks diminishing our ability fully to see and comprehend the former. To speak of the Indian "discovery of the principle of the differential calculus" somewhat obscures the fact that Indian techniques for expressing changes in the Sine by means of the Cosine or vice versa, as in the examples we have seen, remained within that specific trigonometric context. The differential "principle" was not generalized to arbitrary functions—in fact, the explicit notion of an arbitrary function, not to mention that of its derivative or an algorithm for taking the derivative, is irrelevant here"</ref>

In the 17th century, [[James Gregory (astronomer and mathematician)|James Gregory]] worked in the new [[decimal]] system on infinite series and published several [[Maclaurin series]]. In 1715, a general method for constructing the [[Taylor series]] for all functions for which they exist was provided by [[Brook Taylor]]. [[Leonhard Euler]] in the 18th century, developed the theory of [[hypergeometric series]] and [[q-series]].

===Convergence criteria===
The investigation of the validity of infinite series is considered to begin with [[Carl Friedrich Gauss|Gauss]] in the 19th century. Euler had already considered the hypergeometric series

<math display=block>1 + \frac{\alpha\beta}{1\cdot\gamma}x + \frac{\alpha(\alpha+1)\beta(\beta+1)}{1 \cdot 2 \cdot \gamma(\gamma+1)}x^2 + \cdots</math>

on which Gauss published a memoir in 1812. It established simpler criteria of convergence, and the questions of remainders and the range of convergence.

[[Cauchy]] (1821) insisted on strict tests of convergence; he showed that if two series are convergent their product is not necessarily so, and with him begins the discovery of effective criteria. The terms ''convergence'' and ''divergence'' had been introduced long before by [[James Gregory (astronomer and mathematician)|Gregory]] (1668). [[Leonhard Euler]] and [[Carl Friedrich Gauss|Gauss]] had given various criteria, and [[Colin Maclaurin]] had anticipated some of Cauchy's discoveries. Cauchy advanced the theory of [[power series]] by his expansion of a complex [[function (mathematics)|function]] in such a form.

[[Niels Henrik Abel|Abel]] (1826) in his memoir on the [[binomial series]]

<math display=block>1 + \frac{m}{1!}x + \frac{m(m-1)}{2!}x^2 + \cdots</math>

corrected certain of Cauchy's conclusions, and gave a completely scientific summation of the series for complex values of <math>m</math> and <math>x</math>. He showed the necessity of considering the subject of continuity in questions of convergence.

Cauchy's methods led to special rather than general criteria, and
the same may be said of [[Joseph Ludwig Raabe|Raabe]] (1832), who made the first elaborate investigation of the subject, of [[Augustus De Morgan|De Morgan]] (from 1842), whose
logarithmic test [[Paul du Bois-Reymond|DuBois-Reymond]] (1873) and [[Alfred Pringsheim|Pringsheim]] (1889) have
shown to fail within a certain region; of [[Joseph Louis François Bertrand|Bertrand]] (1842), [[Pierre Ossian Bonnet|Bonnet]]
(1843), [[Carl Johan Malmsten|Malmsten]] (1846, 1847, the latter without integration); [[George Gabriel Stokes|Stokes]] (1847), [[Paucker]] (1852), [[Chebyshev]] (1852), and [[Arndt]]
(1853).

General criteria began with [[Ernst Kummer|Kummer]] (1835), and have been studied by [[Gotthold Eisenstein|Eisenstein]] (1847), [[Weierstrass]] in his various
contributions to the theory of functions, [[Ulisse Dini|Dini]] (1867),
DuBois-Reymond (1873), and many others. Pringsheim's memoirs (1889) present the most complete general theory.

===Uniform convergence===
The theory of [[uniform convergence]] was treated by Cauchy (1821), his limitations being pointed out by Abel, but the first to attack it
successfully were [[Philipp Ludwig von Seidel|Seidel]] and [[George Gabriel Stokes|Stokes]] (1847–48). Cauchy took up the
problem again (1853), acknowledging Abel's criticism, and reaching
the same conclusions which Stokes had already found. Thomae used the
doctrine (1866), but there was great delay in recognizing the importance of distinguishing between uniform and non-uniform
convergence, in spite of the demands of the theory of functions.

===Semi-convergence===
A series is said to be semi-convergent (or conditionally convergent) if it is convergent but not [[absolute convergence|absolutely convergent]].

Semi-convergent series were studied by Poisson (1823), who also gave a general form for the remainder of the Maclaurin formula. The most important solution of the problem is due, however, to Jacobi (1834), who attacked the question of the remainder from a different standpoint and reached a different formula. This expression was also worked out, and another one given, by [[Carl Johan Malmsten|Malmsten]] (1847). [[Schlömilch]] (''Zeitschrift'', Vol.I, p.&nbsp;192, 1856) also improved Jacobi's remainder, and showed the relation between the remainder and [[Faulhaber's formula|Bernoulli's function]]

<math display=block>F(x) = 1^n + 2^n + \cdots + (x - 1)^n.</math>

[[Angelo Genocchi|Genocchi]] (1852) has further contributed to the theory.

Among the early writers was [[Josef Hoene-Wronski|Wronski]], whose "loi suprême" (1815) was hardly recognized until [[Arthur Cayley|Cayley]] (1873) brought it into
prominence.

===Fourier series===
[[Fourier series]] were being investigated
as the result of physical considerations at the same time that
Gauss, Abel, and Cauchy were working out the theory of infinite
series. Series for the expansion of sines and cosines, of multiple
arcs in powers of the sine and cosine of the arc had been treated by
[[Jacob Bernoulli]] (1702) and his brother [[Johann Bernoulli]] (1701) and still
earlier by [[Franciscus Vieta|Vieta]]. Euler and [[Joseph Louis Lagrange|Lagrange]] simplified the subject,
as did [[Louis Poinsot|Poinsot]], [[Karl Schröter|Schröter]], [[James Whitbread Lee Glaisher|Glaisher]], and [[Ernst Kummer|Kummer]].

Fourier (1807) set for himself a different problem, to
expand a given function of {{tmath|x}} in terms of the sines or cosines of
multiples of {{tmath|x}}, a problem which he embodied in his ''[[Théorie analytique de la chaleur]]'' (1822). Euler had already given the formulas for determining the coefficients in the series;
Fourier was the first to assert and attempt to prove the general
theorem. [[Siméon Denis Poisson|Poisson]] (1820–23) also attacked the problem from a
different standpoint. Fourier did not, however, settle the question
of convergence of his series, a matter left for [[Augustin Louis Cauchy|Cauchy]] (1826) to
attempt and for Dirichlet (1829) to handle in a thoroughly
scientific manner (see [[convergence of Fourier series]]). Dirichlet's treatment (''[[Journal für die reine und angewandte Mathematik|Crelle]]'', 1829), of trigonometric series was the subject of criticism and improvement by
Riemann (1854), Heine, [[Rudolf Lipschitz|Lipschitz]], [[Ludwig Schläfli|Schläfli]], and
[[Paul du Bois-Reymond|du Bois-Reymond]]. Among other prominent contributors to the theory of
trigonometric and Fourier series were [[Ulisse Dini|Dini]], [[Charles Hermite|Hermite]], [[Georges Henri Halphen|Halphen]],
Krause, Byerly and [[Paul Émile Appell|Appell]].

== Summations over general index sets ==

Definitions may be given for infinitary sums over an arbitrary index set <math>I.</math><ref>{{citation|author=[[Jean Dieudonné]]|title=Foundations of mathematical analysis|publisher=Academic Press}}{{pn|date=August 2025}}</ref>  This generalization introduces two main differences from the usual notion of series: first, there may be no specific order given on the set <math>I</math>; second, the set <math>I</math> may be uncountable. The notions of convergence need to be reconsidered for these, then, because for instance the concept of [[conditional convergence]] depends on the ordering of the index set.

If <math>a : I \mapsto G</math> is a [[Function (mathematics)|function]] from an [[index set]] <math>I</math> to a set <math>G,</math> then the "series" associated to <math>a</math> is the [[formal sum]] of the elements <math>a(x) \in G </math> over the index elements <math>x \in I</math> denoted by the

<math display=block>\sum_{x \in I} a(x).</math>

When the index set is the natural numbers <math>I=\N,</math> the function <math>a : \N \mapsto G</math> is a [[sequence]] denoted by <math>a(n) = a_n.</math> A series indexed on the natural numbers is an ordered formal sum and so we rewrite <math display=inline>\sum_{n \in \N}</math> as <math display=inline>\sum_{n=0}^{\infty}</math> in order to emphasize the ordering induced by the natural numbers. Thus, we obtain the common notation for a series indexed by the natural numbers

<math display=block>\sum_{n=0}^{\infty} a_n = a_0 + a_1 + a_2 + \cdots.</math>

=== Families of non-negative numbers ===
When summing a family <math>\left\{a_i : i \in I\right\}</math> of non-negative real numbers over the index set <math>I</math>, define

<math display=block>\sum_{i\in I}a_i = \sup \biggl\{ \sum_{i\in A} a_i\, : A \subseteq I,  A \text{ finite}\biggr\} \in [0, +\infty].</math>

Any sum over non-negative reals can be understood as the integral of a non-negative function with respect to the [[counting measure]], which accounts for the many similarities between the two constructions.

When the supremum is finite then the set of <math>i \in I</math> such that <math>a_i > 0</math> is countable.  Indeed, for every <math>n \geq 1,</math> the [[cardinality]] <math>\left|A_n\right|</math> of the set <math>A_n = \left\{i \in I : a_i > 1/n\right\}</math> is finite because

<math display=block>\frac{1}{n} \, \left|A_n\right| = \sum_{i \in A_n} \frac{1}{n} \leq \sum_{i \in A_n} a_i \leq \sum_{i \in I} a_i < \infty.</math>

Hence the set <math>A = \left\{i \in I : a_i > 0\right\} = \bigcup_{n = 1}^\infty A_n</math> is [[Countable set|countable]].

If <math>I</math> is countably infinite and enumerated as <math>I = \left\{i_0, i_1, \ldots\right\}</math> then the above defined sum satisfies

<math display=block>\sum_{i \in I} a_i = \sum_{k=0}^{\infty} a_{i_k},</math>
provided the value <math>\infty</math> is allowed for the sum of the series.

=== Abelian topological groups ===
Let <math>a : I \to X</math> be a map, also denoted by <math>\left(a_i\right)_{i \in I},</math> from some non-empty set <math>I</math> into a [[Hausdorff space|Hausdorff]] [[Abelian group|abelian]] [[topological group]] <math>X.</math> 
Let <math>\operatorname{Finite}(I)</math> be the collection of all [[Finite set|finite]] [[subset]]s of <math>I,</math> with <math>\operatorname{Finite}(I)</math> viewed as a [[directed set]], [[Partially ordered set|ordered]] under [[Inclusion (mathematics)|inclusion]] <math>\,\subseteq\,</math> with [[Union (set theory)|union]] as [[Join (mathematics)|join]]. 
The family <math>\left(a_i\right)_{i \in I},</math> is said to be {{em|[[unconditionally summable]]}} if the following [[Limit of a net|limit]], which is denoted by <math>\textstyle \sum_{i\in I} a_i</math> and is called the {{em|sum}} of <math>\left(a_i\right)_{i \in I},</math> exists in <math>X:</math>

<math display=block>\sum_{i\in I} a_i := \lim_{A \in \operatorname{Finite}(I)} \ \sum_{i\in A} a_i = \lim \biggl\{\sum_{i\in A} a_i \,: A \subseteq I, A \text{ finite }\biggr\}</math>
Saying that the sum <math>\textstyle S := \sum_{i\in I} a_i</math> is the limit of finite partial sums means that for every neighborhood <math>V</math> of the origin in <math>X,</math> there exists a finite subset <math>A_0</math> of <math>I</math> such that

<math display=block>S - \sum_{i \in A} a_i \in V \qquad \text{ for every finite superset} \; A \supseteq A_0.</math>

Because <math>\operatorname{Finite}(I)</math> is not [[Total order|totally ordered]], this is not a [[limit of a sequence]] of partial sums, but rather of a [[Net (mathematics)|net]].<ref name="Bourbaki">{{cite book|title=General Topology: Chapters 1–4|first=Nicolas|last=Bourbaki|author-link=Nicolas Bourbaki|year=1998|publisher=Springer|isbn=978-3-540-64241-1|pages=261–270}}</ref><ref name="Choquet">{{cite book|title=Topology|first=Gustave|last=Choquet|author-link=Gustave Choquet|year=1966|publisher=Academic Press|isbn=978-0-12-173450-3|pages=216–231}}</ref>

For every neighborhood <math>W</math> of the origin in <math>X,</math> there is a smaller neighborhood <math>V</math> such that <math>V - V \subseteq W.</math>  It follows that the finite partial sums of an unconditionally summable family <math>\left(a_i\right)_{i \in I},</math> form a {{em|[[Cauchy net]]}}, that is, for every neighborhood <math>W</math> of the origin in <math>X,</math> there exists a finite subset <math>A_0</math> of <math>I</math> such that

<math display=block>\sum_{i \in A_1} a_i - \sum_{i \in A_2} a_i \in W \qquad \text{ for all finite supersets } \; A_1, A_2 \supseteq A_0,</math>
which implies that <math>a_i \in W</math> for every <math>i \in I \setminus A_0</math> (by taking <math>A_1 := A_0 \cup \{i\}</math> and <math>A_2 := A_0</math>).

When <math>X</math> is [[Complete topological group|complete]], a family <math>\left(a_i\right)_{i \in I}</math> is unconditionally summable in <math>X</math> if and only if the finite sums satisfy the latter Cauchy net condition. When <math>X</math> is complete and <math>\left(a_i\right)_{i \in I},</math> is unconditionally summable in <math>X,</math> then for every subset <math>J \subseteq I,</math> the corresponding subfamily <math>\left(a_j\right)_{j \in J},</math> is also unconditionally summable in <math>X.</math>

When the sum of a family of non-negative numbers, in the extended sense defined before, is finite, then it coincides with the sum in the topological group <math>X = \R.</math>

If a family <math>\left(a_i\right)_{i \in I}</math> in <math>X</math> is unconditionally summable then for every neighborhood <math>W</math> of the origin in <math>X,</math> there is a finite subset <math>A_0 \subseteq I</math> such that <math>a_i \in W</math> for every index <math>i</math> not in <math>A_0.</math>  If <math>X</math> is a [[first-countable space]] then it follows that the set of <math>i \in I</math> such that <math>a_i \neq 0</math> is countable.  This need not be true in a general abelian topological group (see examples below).

=== Unconditionally convergent series ===
Suppose that <math>I = \N.</math>  If a family <math>a_n, n \in \N,</math> is unconditionally summable in a Hausdorff [[abelian topological group]] <math>X,</math> then the series in the usual sense converges and has the same sum,

<math display=block>\sum_{n=0}^\infty a_n = \sum_{n \in \N} a_n.</math>

By nature, the definition of unconditional summability is insensitive to the order of the summation.  When <math>\textstyle \sum a_n</math> is unconditionally summable, then the series remains convergent after any [[permutation]] <math>\sigma : \N \to \N</math> of the set <math>\N</math> of indices, with the same sum,

<math display=block>\sum_{n=0}^\infty a_{\sigma(n)} = \sum_{n=0}^\infty a_n.</math>

Conversely, if every permutation of a series <math>\textstyle \sum a_n</math> converges, then the series is unconditionally convergent.  When <math>X</math> is [[Complete topological group|complete]] then unconditional convergence is also equivalent to the fact that all subseries are convergent; if <math>X</math> is a [[Banach space]], this is equivalent to say that for every sequence of signs <math>\varepsilon_n = \pm 1</math><!-- this is not about convergence of functions, even less about uniform convergence. -->, the series

<math display=block>\sum_{n=0}^\infty \varepsilon_n a_n</math>

converges in <math>X.</math>

=== Series in topological vector spaces ===
If <math>X</math> is a [[topological vector space]] (TVS) and <math>\left(x_i\right)_{i \in I}</math> is a (possibly [[uncountable]]) family in <math>X</math> then this family is '''summable'''<ref>{{Cite book |last1=Schaefer |first1=Helmut H. |author-link1=Helmut H. Schaefer |title=Topological Vector Spaces |last2=Wolff |first2=Manfred P. |year=1999 |publisher=Springer |isbn=978-1-4612-7155-0 |edition=2nd |series=Graduate Texts in Mathematics |volume=8 |location=New York, NY |pages=179–180}}</ref> if the limit <math>\textstyle \lim_{A \in \operatorname{Finite}(I)} x_A</math> of the [[Net (mathematics)|net]] <math>\left(x_A\right)_{A \in \operatorname{Finite}(I)}</math> exists in <math>X,</math> where <math>\operatorname{Finite}(I)</math> is the [[directed set]] of all finite subsets of <math>I</math> directed by inclusion <math>\,\subseteq\,</math> and <math display=inline>x_A := \sum_{i \in A} x_i.</math>

It is called '''[[absolutely summable]]''' if in addition, for every continuous seminorm <math>p</math> on <math>X,</math> the family <math>\left(p\left(x_i\right)\right)_{i \in I}</math> is summable.
If <math>X</math> is a normable space and if <math>\left(x_i\right)_{i \in I}</math> is an absolutely summable family in <math>X,</math> then necessarily all but a countable collection of <math>x_i</math>'s are zero. Hence, in normed spaces, it is usually only ever necessary to consider series with countably many terms.

Summable families play an important role in the theory of [[nuclear space]]s.

=== Series in Banach and seminormed spaces ===
The notion of series can be easily extended to the case of a [[seminormed space]]. 
If <math>x_n</math> is a sequence of elements of a normed space <math>X</math> and if <math>x \in X</math> then the series <math>\textstyle \sum x_n</math> converges to <math>x</math> in <math>X</math> if the sequence of partial sums of the series <math display=inline>\bigl(\!\!~\sum_{n=0}^N x_n\bigr)_{N=1}^{\infty}</math> converges to <math>x</math> in <math>X</math>; to wit,

<math display=block>\Biggl\|x - \sum_{n=0}^N x_n\Biggr\| \to 0 \quad \text{ as } N \to \infty.</math>

More generally, convergence of series can be defined in any [[Abelian group|abelian]] [[Hausdorff space|Hausdorff]] [[topological group]]. 
Specifically, in this case, <math>\textstyle \sum x_n</math> converges to <math>x</math> if the sequence of partial sums converges to <math>x.</math>

If <math>(X, |\cdot|)</math> is a [[seminormed space]], then the notion of absolute convergence becomes: 
A series <math display=inline>\sum_{i \in I} x_i</math> of vectors in <math>X</math> '''converges absolutely''' if

<math display=block> \sum_{i \in I} \left|x_i\right| < +\infty</math>

in which case all but at most countably many of the values <math>\left|x_i\right|</math> are necessarily zero.

If a countable series of vectors in a Banach space converges absolutely then it converges unconditionally, but the converse only holds in finite-dimensional Banach spaces (theorem of {{harvtxt|Dvoretzky|Rogers|1950}}).

=== Well-ordered sums ===
Conditionally convergent series can be considered if <math>I</math> is a [[well-ordered]] set, for example, an [[ordinal number]] <math>\alpha_0.</math>
In this case, define by [[transfinite recursion]]:

<math display=block>\sum_{\beta < \alpha + 1}\! a_\beta = a_{\alpha} + \sum_{\beta < \alpha} a_\beta</math>

and for a limit ordinal <math>\alpha,</math>

<math display=block>\sum_{\beta < \alpha} a_\beta = \lim_{\gamma\to\alpha}\, \sum_{\beta < \gamma} a_\beta</math>

if this limit exists.  If all limits exist up to <math>\alpha_0,</math> then the series converges.

=== Examples ===
* Given a function <math>f : X \to Y</math> into an abelian topological group <math>Y,</math> define for every <math>a \in X,</math> <math display=block>
f_a(x)= \begin{cases}
0 & x\neq a, \\
f(a) & x=a, \\
\end{cases}</math> a function whose [[Support (mathematics)|support]] is a [[Singleton (mathematics)|singleton]] <math>\{a\}.</math> Then <math display=block>f = \sum_{a \in X}f_a</math> in the [[topology of pointwise convergence]] (that is, the sum is taken in the infinite product group <math>\textstyle Y^{X}</math>).
* In the definition of [[partitions of unity]], one constructs sums of functions over arbitrary index set <math>I,</math> <math display=block>
\sum_{i \in I} \varphi_i(x) = 1.
</math> While, formally, this requires a notion of sums of uncountable series, by construction there are, for every given <math>x,</math> only finitely many nonzero terms in the sum, so issues regarding convergence of such sums do not arise.  Actually, one usually assumes more: the family of functions is ''locally finite'', that is, for every <math>x</math> there is a neighborhood of <math>x</math> in which all but a finite number of functions vanish.  Any regularity property of the <math>\varphi_i,</math> such as continuity, differentiability, that is preserved under finite sums will be preserved for the sum of any subcollection of this family of functions.
* On the [[first uncountable ordinal]] <math>\omega_1</math> viewed as a topological space in the [[order topology]], the constant function <math>f : \left[0, \omega_1\right) \to \left[0, \omega_1\right]</math> given by <math>f(\alpha) = 1</math> satisfies <math display=block>
\sum_{\alpha \in [0,\omega_1)}\!\!\! f(\alpha) = \omega_1
</math> (in other words, <math>\omega_1</math> copies of 1 is <math>\omega_1</math>) only if one takes a limit over all ''countable'' partial sums, rather than finite partial sums. This space is not separable.

==See also==
* [[Continued fraction]]
* [[Convergence tests]]
* [[Convergent series]]
* [[Divergent series]]
* [[Infinite compositions of analytic functions]]
* [[Infinite expression (mathematics)|Infinite expression]]
* [[Infinite product]]
* [[Iterated binary operation]]
* [[List of mathematical series]]
* [[Prefix sum]]
* [[Sequence transformation]]
* [[Series expansion]]

==Notes==
{{Reflist}}

== References ==

* {{Cite book
  | last = Apostol | first = Tom M. | author-link = Tom M. Apostol
  | year = 1967 |orig-year=1961
  | title = Calculus | title-link = Calculus (Apostol books)
  | volume = 1
  | edition = 2nd
  | publisher = John Wiley & Sons
  | isbn = 0-471-00005-1
}}
* {{Cite book
  | last = Rudin | first = Walter | author-link = Walter Rudin
  | year = 1976 | orig-date = 1953
  | title = [[Principles of Mathematical Analysis]]
  | edition = 3rd
  | location = New York 
  | publisher = McGraw-Hill
  | isbn = 0-07-054235-X
  | oclc = 1502474 }}
* {{Cite book
  |last=Spivak |first=Michael |author-link=Michael Spivak
  |year=2008 |orig-year=1967
  |title=Calculus
  |edition=4th
  |location=Houston, TX |publisher=Publish or Perish
  |isbn=978-0-914098-91-1
  }}

==Further reading==
* {{cite book
  | last = Bromwich | first = T. J. | author-link = Thomas John I'Anson Bromwich
  | year = 1926
  | title = An Introduction to the Theory of Infinite Series
  | edition = 2nd
  | publisher = MacMillan
  }}
* {{cite journal |doi=10.1073/pnas.36.3.192 |last1=Dvoretzky |first1=Aryeh |last2=Rogers |first2=C.&nbsp;Ambrose |title=Absolute and unconditional convergence in normed linear spaces |journal=Proc. Natl. Acad. Sci. U.S.A. |volume=36 |issue=3 |year=1950 |pages=192–197 |pmid=16588972 |pmc=1063182 |bibcode=1950PNAS...36..192D |doi-access=free}}
* {{cite book
  | last1 = Narici | first1 = Lawrence
  | last2 = Beckenstein | first2 = Edward
  | year = 2011
  | title = Topological Vector Spaces
  | edition = 2nd
  | location = Boca Raton, FL | publisher = CRC Press
  | isbn = 978-1584888666
  }}
* {{citation |last=Swokowski |first=Earl W. |title=Calculus with analytic geometry |edition=Alternate |year=1983 |publisher=Prindle, Weber & Schmidt |place=Boston |isbn=978-0-87150-341-1 |url=https://archive.org/details/calculuswithanal00swok}}
* {{cite book |last=Pietsch |first=Albrecht |title=Nuclear locally convex spaces |publisher=Springer-Verlag |publication-place=Berlin, New York |year=1972 |isbn=0-387-05644-0 |oclc=539541}} 
* {{cite book |last=Robertson |first=A. P. |title=Topological vector spaces |publisher=University Press |publication-place=Cambridge England |year=1973 |isbn=0-521-29882-2 }}
* {{cite book |last=Ryan |first=Raymond |title=Introduction to tensor products of Banach spaces |publisher=Springer |publication-place=London New York |year=2002 |isbn=1-85233-437-1 |oclc=48092184}}
* {{cite book
  | last1 = Schaefer | first1 = Helmut H. | author-link1 = Helmut H. Schaefer
  | last2 = Wolff | first2 = Manfred P.
  | year = 1999
  | title = Topological Vector Spaces
  | edition = 2nd
  | location = New York | publisher = Springer
  | isbn = 978-1-4612-7155-0
  }}
* {{cite book
  | last = Trèves | first = François | author-link = François Trèves
  | year = 1967
  | title = Topological Vector Spaces, Distributions and Kernels
  | location = New York | publisher = Academic Press
  }} Reprinted by Dover, 2006, {{isbn|978-0-486-45352-1}}.
* {{cite book |last=Wong |title=Schwartz spaces, nuclear spaces, and tensor products |publisher=Springer-Verlag |publication-place=Berlin New York |year=1979 |isbn=3-540-09513-6 |oclc=5126158}}

==External links==
{{Commons category|Series (mathematics)}}
* {{SpringerEOM |title=Series |id=p/s084670 }}
* [http://www.math.odu.edu/~bogacki/citat/series/index.html Infinite Series Tutorial]
* {{cite web |url=http://tutorial.math.lamar.edu/Classes/CalcII/Series_Basics.aspx |title=Series-TheBasics |publisher=Paul's Online Math Notes}}
* {{cite web |url=http://lesliegreen.byethost3.com/articles/series.pdf |title=Show-Me Collection of Series |publisher=Leslie Green}}

{{Series (mathematics)}}
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{{DEFAULTSORT:Series (Mathematics)}}
[[Category:Calculus]]
[[Category:Series (mathematics)| ]]