View source for Series (mathematics)

{{Short description|Infinite sum}}
{{Calculus}}

In [[Mathematics|mathematics]], a '''series''' is, roughly speaking, the operation of [[Addition|adding]] infinitely many quantities, one after the other, to a given starting quantity.<ref>{{cite book |title=Calculus Made Easy |last1=Thompson |first1=Silvanus |last2=Gardner |first2=Martin |author-link2=Martin Gardner |date=1998 |isbn=978-0-312-18548-0 |url=https://archive.org/details/calculusmadeeasy00thom_0 }}</ref>  The study of series is a major part of [[Calculus|calculus]] and its generalization, [[Mathematical analysis|mathematical analysis]]. Series are used in most areas of mathematics, even for studying finite structures (such as in [[Combinatorics|combinatorics]]) through [[Generating function|generating function]]s. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as [[Physics:Physics|physics]], [[Computer science|computer science]], [[Statistics|statistics]] and [[Finance:Finance|finance]].

For a long time, the idea that such a potentially infinite [[Summation|summation]] could produce a finite result was considered [[Philosophy:Paradox|paradox]]ical. This paradox was resolved using the concept of a [[Limit (mathematics)|limit]] during the 17th century. Zeno's paradox of Achilles and the tortoise illustrates this counterintuitive property of infinite sums: 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. [[Biography:Zeno of Elea|Zeno]] concluded that Achilles could ''never'' reach the tortoise, and thus that movement does not exist. 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 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.

In modern terminology, any (ordered) infinite sequence <math>(a_1,a_2,a_3,\ldots)</math> of terms (that is, numbers, [[Function (mathematics)|functions]], or anything that can be added) defines a series, which is the operation of adding the {{math|''a''<sub>''i''</sub>}} one after the other. To emphasize that there are an infinite number of terms, a series may be called an '''infinite series'''. Such a series is represented (or denoted) by an [[Expression (mathematics)|expression]] like

<math display=block>a_1+a_2+a_3+\cdots,</math>

or, using the summation sign,

<math display=block>\sum_{i=1}^\infty a_i.</math>

The infinite sequence of additions implied by a series cannot be effectively carried on (at least in a finite amount of time). However, if the [[Set (mathematics)|set]] to which the terms and their finite sums belong has a notion of [[Limit (mathematics)|limit]], it is sometimes possible to assign a value to a series, called the sum of the series. This value is the limit as {{math|''n''}} tends to [[Infinity|infinity]] (if the limit exists) of the finite sums of the {{math|''n''}} first terms of the series, which are called the {{math|''n''}}th '''partial sums''' of the series. That is,

<math display=block>\sum_{i=1}^\infty a_i  = \lim_{n\to\infty} \sum_{i=1}^n a_i.</math>

When this limit exists, one says that the series is '''convergent''' or '''summable''', or that the sequence <math>(a_1,a_2,a_3,\ldots)</math> is '''summable'''. In this case, the limit is called the '''sum''' of the series. Otherwise, the series is said to be '''divergent'''.<ref name=":0">{{Cite web|last=Weisstein|first=Eric W.|title=Series|url=https://mathworld.wolfram.com/Series.html|access-date=2020-08-30|website=mathworld.wolfram.com|language=en}}</ref>

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

Generally, the terms of a series come from a [[Ring (mathematics)|ring]], often the [[Field (mathematics)|field]] <math>\mathbb R</math> of the [[Real number|real number]]s or the field <math>\mathbb C</math> of the [[Complex number|complex number]]s. In this case, the set of all series is itself a ring (and even an [[Associative algebra|associative algebra]]), in which the addition consists of adding the series term by term, and the multiplication is the [[Cauchy product]].

==Basic properties==

An infinite series or simply a series is an infinite sum, represented by an [[Infinite expression|infinite expression]] of the form<ref name="SW501">{{harvnb|Swokowski|1983|loc=p. 501}}</ref>

<math display=block>a_0 + a_1 + a_2 + \cdots, </math>

where <math>(a_n)</math> is any ordered [[Sequence|sequence]] of terms, such as [[Number|number]]s, [[Function (mathematics)|functions]], or anything else that can be [[Addition|added]] (an [[Abelian group|abelian group]]).  This is an expression that is obtained from the list of terms <math>a_0,a_1,\dots</math> by laying them side by side, and conjoining them with the symbol "+".  A series may also be represented by using [[Summation#Capital-sigma notation|summation notation]], such as

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

If an abelian group {{math|''A''}} of terms has a concept of [[Limit (mathematics)|limit]] (e.g., if it is a [[Metric space|metric space]]), then some series, the [[Convergent series|convergent series]], can be interpreted as having a value in {{math|''A''}}, called the ''sum of the series''. This includes the common cases from [[Calculus|calculus]], in which the group is the field of [[Real number|real number]]s or the field of [[Complex number|complex number]]s. Given a series <math display="inline">s=\sum_{n=0}^\infty a_n</math>, its {{math|''k''}}th '''partial sum''' is<ref name=":0" />

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

By definition, the series <math display="inline">\sum_{n=0}^{\infty} a_n</math> ''converges'' to the limit {{math|''L''}} (or simply ''sums'' to {{math|''L''}}), if the sequence of its partial sums has a limit {{math|''L''}}.<ref name="SW501" /> In this case, one usually writes

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

A series is said to be ''convergent'' if it converges to some limit, or ''divergent'' when it does not. The value of this limit, if it exists, is then the value of the series.

===Convergent series===
[[File:Geometric sequences.svg|thumb|right|Illustration of 3 [[Geometric series|geometric series]] with partial sums from 1 to 6 terms. The dashed line represents the limit.]]
A series {{math|Σ''a''<sub>''n''</sub>}} is said to [[Convergent series|converge]] or to ''be convergent'' when the sequence {{math|(''s''<sub>''k''</sub>)}} of partial sums has a finite [[Limit of a sequence|limit]]. If the limit of {{math|''s''<sub>''k''</sub>}} is infinite or does not exist, the series is said to [[Divergent series|diverge]].<ref>{{citation|title=Calculus|author=Michael Spivak}}</ref><ref name=":0" /> When the limit of partial sums exists, it is called the value (or sum) of the series

<math display=block>\sum_{n=0}^\infty a_n = \lim_{k\to\infty} s_k = \lim_{k\to\infty} \sum_{n=0}^k a_n.</math>

An easy way that an infinite series can converge is if all the {{math|''a''<sub>''n''</sub>}} are zero for {{math|''n''}} sufficiently large.  Such a series can be identified with a finite sum, so it is only infinite in a trivial sense.

Working out the properties of the series that converge, even if infinitely many terms are nonzero, is the essence of the study of series. Consider the example

<math display=block> 1 + \frac{1}{2}+ \frac{1}{4}+ \frac{1}{8}+\cdots+ \frac{1}{2^n}+\cdots.</math>

It is possible to "visualize" its convergence on the [[Real number|real number line]]: we can imagine a [[Line (geometry)|line]] of length 2, with successive [[Line segment|segment]]s marked off of lengths 1, 1/2, 1/4, etc. There is always room to mark the next segment, because the amount of line remaining is always the same as the last segment marked: When we have marked off 1/2, we still have a piece of length 1/2 unmarked, so we can certainly mark the next&nbsp;1/4. This argument does not prove that the sum is ''equal'' to 2 (although it is), but it does prove that it is ''at most''&nbsp;2.  In other words, the series has an upper bound. Given that the series converges, proving that it is equal to 2 requires only [[Elementary algebra|elementary algebra]]. If the series is denoted {{math|''S''}}, it can be seen that

<math display=block>S/2 = \frac{1+ \frac{1}{2}+ \frac{1}{4}+ \frac{1}{8}+\cdots}{2} = \frac{1}{2}+ \frac{1}{4}+ \frac{1}{8}+ \frac{1}{16} +\cdots.</math>

Therefore,

<math display=block>S-S/2 = 1 \Rightarrow S = 2.</math>

The idiom can be extended to other, equivalent notions of series. For instance, a [[Repeating decimal|recurring decimal]], as in

<math display=block>x = 0.111\dots , </math>

encodes the series

<math display=block>\sum_{n=1}^\infty \frac{1}{10^n}.</math>

Since these series always converge to real numbers (because of what is called the completeness property of the real numbers), to talk about the series in this way is the same as to talk about the numbers for which they stand. In particular, the decimal expansion 0.111... can be identified with 1/9. This leads to an argument that {{nowrap|1=9 × 0.111... = 0.999... = 1}}, which only relies on the fact that the limit laws for series preserve the arithmetic operations; for more detail on this argument, see [[0.999...|0.999...]].

==Examples of numerical series==
* A ''[[Geometric series|geometric series]]'' is one where each successive term is produced by multiplying the previous term by a [[Mathematical constant|constant number]] (called the common ratio in this context). For example:<ref name=":0" /> <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><p>In general, the geometric series</p> <math display=block>\sum_{n=0}^\infty z^n</math> <p>converges [[If and only if|if and only if]] <math display="inline">|z| < 1</math>, in which case it converges to <math display="inline> {1 \over 1 - z}</math>.</p>
* The ''[[Harmonic series (mathematics)|harmonic series]]'' is the series<ref>{{Cite web|title=Infinite Series|url=https://www.mathsisfun.com/algebra/infinite-series.html|access-date=2020-08-30|website=www.mathsisfun.com}}</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> <p>The harmonic series is [[Harmonic series (mathematics)#Divergence|divergent]].</p>
* An ''[[Alternating series|alternating series]]'' is a series where terms alternate signs.  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) \quad </math> <p>(alternating harmonic series) and</p> <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>
* A [[Telescoping series|telescoping series]] <math display=block>\sum_{n=1}^\infty (b_n-b_{n+1})</math> <p>converges if the [[Sequence|sequence]] ''b''<sub>''n''</sub> converges to a limit ''L''—as ''n'' goes to infinity. The value of the series is then ''b''<sub>1</sub> − ''L''.</p>
* An ''arithmetico-geometric series'' is a generalization of the geometric series, which has [[Coefficient|coefficient]]s of the common ratio equal to the terms in an arithmetic sequence.  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 [[Harmonic series (mathematics)#P-series|''p''-series]] <math display=block>\sum_{n=1}^\infty\frac{1}{n^p}</math> <p>converges for ''p'' > 1 and diverges for ''p'' ≤ 1, which can be shown with the integral criterion described below in [[Series (mathematics)#Convergence tests|convergence tests]]. As a function of ''p'', the sum of this series is [[Riemann zeta function|Riemann's zeta function]].</p>
* Hypergeometric series: <math display=block>_rF_s \left[ \begin{matrix}a_1, a_2, \dotsc, a_r \\ b_1, b_2, \dotsc, b_s \end{matrix}; z \right] := \sum_{n=0}^{\infty} \frac{(a_1)_n (a_2)_n \dotsb (a_r)_n}{(b_1)_n (b_2)_n \dotsb (b_s)_n \; n!} z^n</math> <p>and their generalizations (such as [[Basic hypergeometric series|basic hypergeometric series]] and [[Elliptic hypergeometric series|elliptic hypergeometric series]]) frequently appear in [[Physics:Integrable systems|integrable systems]] and [[Physics:Mathematical physics|mathematical physics]].<ref>Gasper, G., Rahman, M. (2004). Basic hypergeometric series. {{wipe|Cambridge University Press}}.</ref></p>
* There are some elementary series whose convergence is not yet known/proven. For example, it is unknown whether the Flint Hills series {{anchor|Flint Hills series|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 ''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 ''n'' that are close to <math>m\pi</math> for some integer ''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_{i=1}^{\infty} \frac{1}{i^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> \sum_{i=1}^\infty \frac{(-1)^{i+1}(4)}{2i-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===

<math display=block>\sum_{i=1}^\infty \frac{(-1)^{i+1}}{i} = \ln 2</math><ref name=":0" />

<math display=block>\sum_{i=0}^\infty \frac{1}{(2i+1)(2i+2)} = \ln 2</math>

<math display=block>\sum_{i=0}^\infty \frac{(-1)^i}{(i+1)(i+2)} = 2\ln(2) -1</math>

<math display=block>\sum_{i=1}^\infty \frac{1}{i \left(4i^2-1\right)} = 2\ln(2) -1</math>

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

<math display=block> \sum_{i=1}^\infty \left(\frac{1}{3^i}+\frac{1}{4^i}\right)\frac{1}{i} = \ln 2</math>

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

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

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

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

==Calculus and partial summation as an operation on sequences==
Partial summation takes as input a sequence, (''a''<sub>''n''</sub>), and gives as output another sequence, (''S''<sub>''N''</sub>). It is thus a [[Unary operation|unary operation]] on sequences. Further, this function is [[Linear map|linear]], and thus is a linear operator on the [[Vector space|vector space]] of sequences, denoted Σ. The inverse operator is the [[Finite difference|finite difference]] operator, denoted Δ. These behave as discrete analogues of [[Integral|integration]] and [[Derivative|differentiation]], only for series (functions of a natural number) instead of functions of a real variable. For example, the sequence (1, 1, 1, ...) has series (1, 2, 3, 4, ...) as its partial summation, which is analogous to the fact that <math display="inline">\int_0^x 1\,dt = x.</math>

In [[Computer science|computer science]], it is known as [[Prefix sum|prefix sum]].

==Properties of series==
Series are classified not only by whether they converge or diverge, but also by the properties of the terms a<sub>n</sub> (absolute or conditional convergence); type of convergence of the series (pointwise, uniform); the class of the term a<sub>n</sub> (whether it is a real number, arithmetic progression, trigonometric function); etc.

===Non-negative terms===
When ''a<sub>n</sub>'' is a non-negative real number for every ''n'', the sequence ''S<sub>N</sub>'' of partial sums is non-decreasing. It follows that a series Σ''a<sub>n</sub>'' with non-negative terms converges if and only if the sequence ''S<sub>N</sub>'' of partial sums is bounded.

For example, the series

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

is convergent, because the inequality

<math display=block>\frac1 {n^2} \le \frac{1}{n-1} - \frac{1}{n}, \quad n \ge 2,</math>

and a telescopic sum argument implies that the partial sums are bounded by 2. The exact value of the original series is the [[Basel problem]].

===Grouping===
When you group a series reordering of the series does not happen, so [[Riemann series theorem]] does not apply. A new series will have its partial sums as subsequence of original series, which means if the original series converges, so does the new series. But for divergent series that is not true, for example 1-1+1-1+... grouped every two elements will create 0+0+0+... series, which is convergent. On the other hand, divergence of the new series means the original series can be only divergent which is sometimes useful, like in [[Harmonic series (mathematics)#Comparison test|Oresme proof]].

===Absolute convergence===
{{Main|Absolute convergence}}
A series

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

''converges absolutely'' if the series of [[Absolute value|absolute value]]s

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

converges. This is sufficient to guarantee not only that the original series converges to a limit, but also that any reordering of it converges to the same limit.

===Conditional convergence===
{{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. A famous example is the alternating 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 (and its sum is equal to&nbsp;<math>\ln 2</math>), but the series formed by taking the absolute value of each term is the divergent [[Harmonic series (mathematics)|harmonic series]]. The [[Riemann series theorem]] says that any conditionally convergent series can be reordered to make a divergent series, and moreover, if the <math>a_{n}</math> are real and <math>S</math> is any real number, that one can find a reordering so that the reordered series converges with sum equal to&nbsp;<math>S</math>.

[[Abel's test]] is an important tool for handling semi-convergent series. If a series has the form

<math display=block>\sum a_n = \sum \lambda_n b_n</math>

where the partial sums <math>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:

<math display=block>\sup_N \left| \sum_{n=0}^N b_n \right| < \infty, \ \ \sum \left|\lambda_{n+1} - \lambda_n\right| < \infty\ \text{and} \ \lambda_n B_n \ \text{converges,}</math>

then the series <math display="inline">\sum a_{n}</math> is convergent. This applies to the point-wise convergence of many trigonometric series, as in

<math display=block>\sum_{n=2}^\infty \frac{\sin(n x)}{\ln n}</math>

with <math>0 < x < 2\pi</math>.  Abel's method consists in writing <math>b_{n+1}=B_{n+1}-B_{n}</math>, and in performing a transformation similar to [[Integration by parts|integration by parts]] (called [[Summation by parts|summation by parts]]), that relates the given series <math display="inline">\sum a_{n}</math> to the absolutely convergent series

<math display=block> \sum (\lambda_n - \lambda_{n+1}) \, B_n.</math>

===Evaluation of truncation errors===
The evaluation of truncation errors is an important procedure in [[Numerical analysis|numerical analysis]] (especially [[Validated numerics|validated numerics]] and [[Computer-assisted proof|computer-assisted proof]]).

====[[Alternating series]]====
When conditions of the [[Alternating series test|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>

====[[Taylor series]]====
[[Taylor's theorem]] is a statement that includes the evaluation of the error term when the [[Taylor series]] is truncated.

====Hypergeometric series====
By using the [[Ratio|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]]====
For the [[Matrix exponential|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) := \left[\sum_{j=0}^r\frac{1}{j!}(X/s)^j\right]^s,\quad \|\exp(X)-T_{r,s}(X)\|\leq\frac{\|X\|^{r+1}}{s^r(r+1)!}\exp(\|X\|).</math>

==Convergence tests==
{{Main|Convergence tests}}
There exist many tests that can be used to determine whether particular series converge or diverge.
* ''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.
* Comparison test 1 (see [[Direct comparison test]]): 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 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 (though it could still be conditionally convergent, for example, if the <math>a_n</math> alternate in sign).
* Comparison test 2 (see [[Limit comparison test]]): If <math display="inline">\sum b_n</math> is an absolutely convergent series such that <math>\left\vert \frac{a_{n+1}}{a_{n}} \right\vert \leq \left\vert \frac{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 \frac{a_{n+1}}{a_{n}} \right\vert \geq \left\vert \frac{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, for example, if the <math>a_n</math> alternate in sign).
* [[Ratio test]]: If there exists a constant <math>C < 1</math> such that <math>\left\vert \frac{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.
* [[Root test]]: If there exists a constant <math>C < 1</math> such that <math>\left\vert a_{n} \right\vert^{\frac{1}{n}} \leq C</math> for all sufficiently large&nbsp;<math>n</math>, then <math display="inline">\sum a_{n}</math> converges absolutely.
* [[Integral test for convergence|Integral test]]: 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. --> with <math>f(n)=a_{n}</math> for all&nbsp;<math>n</math>, then <math display="inline">\sum a_{n}</math> converges if and only if the [[Integral|integral]] <math display="inline">\int_{1}^{\infty} f(x) \, dx</math> is finite.
* Cauchy's condensation test: If <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 of the same nature: both convergent, or both divergent.
* [[Alternating series test]]: A series of the form <math display="inline">\sum (-1)^{n} a_{n}</math> (with <math>a_{n} > 0</math>) is called ''alternating''. Such a series converges if the [[Sequence|sequence]] ''<math>a_{n}</math>'' is monotone decreasing and converges to&nbsp;<math>0</math>. The converse is in general not true.
* For some specific types of series there are more specialized convergence tests, for instance for [[Fourier series]] there is the [[Dini test]].

==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>

[[Pointwise convergence|converges pointwise]] on a set ''E'', if the series converges for each ''x'' in ''E'' as an ordinary 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 ''ƒ''(''x'') as ''N''&nbsp;→&nbsp;∞ for each ''x''&nbsp;∈&nbsp;''E''.

A stronger notion of convergence of a series of functions is the [[Uniform convergence|uniform convergence]]. A series converges uniformly if it converges pointwise to the function ''ƒ''(''x''), and the error in approximating the limit by the ''N''th partial sum,

<math display=block>|s_N(x) - f(x)|</math>

can be made minimal ''independently'' of ''x'' by choosing a sufficiently large ''N''.

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 ''ƒ''<sub>''n''</sub> are [[Integral|integrable]] on a closed and bounded interval ''I'' and converge uniformly, then the series is also integrable on ''I'' and can be integrated term-by-term. Tests for uniform convergence include the [[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|almost everywhere]] if it converges pointwise except on a certain set of [[Null set|measure zero]].  Other [[Modes of convergence|modes of convergence]] depend on a different [[Metric space|metric space]] structure on the space of functions under consideration.  For instance, a series of functions '''converges in mean''' on a set ''E'' to a limit function ''ƒ'' provided

<math display=block>\int_E \left|s_N(x)-f(x)\right|^2\,dx \to 0</math>

as ''N''&nbsp;→&nbsp;∞.

===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 ''c'' of a function is a power series that, in many cases, converges to the function in a neighborhood of ''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 ''x''.

Unless it converges only at ''x''=''c'', such a series converges on a certain open disc of convergence centered at the point ''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|radius of convergence]], and can in principle be determined from the asymptotics of the coefficients ''a''<sub>''n''</sub>.  The convergence is uniform on [[Closed set|closed]] and [[Bounded set|bounded]] (that is, 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 [[Biography:Leonhard Euler|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|combinatorics]] to describe and study [[Sequence|sequence]]s that are otherwise difficult to handle, for example, using the method of [[Generating function|generating function]]s.  The [[Hilbert–Poincaré series]] is a formal power series used to study graded algebras.

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|addition]], [[Multiplication|multiplication]], [[Derivative|derivative]], [[Antiderivative|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|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|total algebra]] of the [[Monoid|monoid]] of natural numbers over the underlying term ring.<ref>{{citation|author=Nicolas Bourbaki|title=Algebra|publisher=Springer|year=1989}}: §III.2.11.</ref>  If the underlying term ring is a [[Differential algebra|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 ''s'' is a [[Complex number|complex number]].  For example, if all ''a''<sub>''n''</sub> are equal to 1, then 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|analytic number theory]].  Generally a Dirichlet series converges if the real part of ''s'' is greater than a number called the abscissa of convergence.  In many cases, a Dirichlet series can be extended to an [[Analytic function|analytic function]] outside the domain of convergence by [[Analytic continuation|analytic continuation]].  For example, the Dirichlet series for the zeta function converges absolutely when Re(''s'')&nbsp;> 1, but the zeta function can be extended to a holomorphic function defined on <math>\Complex\setminus\{1\}</math> with a simple pole at&nbsp;1.

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

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

<math display=block>\frac12 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.

==History of the theory of infinite series==

===Development of infinite series===
Greek mathematician [[Biography:Archimedes|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|method of exhaustion]] to calculate the [[Area|area]] under the arc of a [[Parabola|parabola]] with the summation of an infinite series, 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 = [[Organization:University of St Andrews|University of St Andrews]]| url = http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/The_rise_of_calculus.html |date=February 1996|access-date= 2007-08-07}}</ref><ref>{{cite journal|url=http://eric.ed.gov/ERICWebPortal/custom/portlets/recordDetails/detailmini.jsp?_nfpb=true&_&ERICExtSearch_SearchValue_0=EJ502088&ERICExtSearch_SearchType_0=no&accno=EJ502088|title=Archimedes and Pi-Revisited.|first=Bidwell, James|last=K.|date=30 November 1993|journal=School Science and Mathematics|volume=94|issue=3}}</ref>

Mathematicians from the [[Astronomy:Kerala school of astronomy and mathematics|Kerala school]] were studying infinite series {{circa|1350 CE}}.<ref>{{cite web|url=http://www.manchester.ac.uk/discover/news/article/?id=2962|title=Indians predated Newton 'discovery' by 250 years|website=manchester.ac.uk}}</ref>

In the 17th century, James Gregory worked in the new [[Decimal|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 [[Biography:Brook Taylor|Brook Taylor]]. [[Biography:Leonhard Euler|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 [[Biography: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 Gregory (1668). [[Biography:Leonhard Euler|Leonhard Euler]] and [[Biography: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|power series]] by his expansion of a complex [[Function (mathematics)|function]] in such a form.

[[Biography:Niels Henrik Abel|Abel]] (1826) in his memoir on the [[Binomial series|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 Raabe (1832), who made the first elaborate investigation of the subject, of [[Biography:Augustus De Morgan|De Morgan]] (from 1842), whose
logarithmic test DuBois-Reymond (1873) and [[Biography:Alfred Pringsheim|Pringsheim]] (1889) have
shown to fail within a certain region; of Bertrand (1842), [[Biography:Pierre Ossian Bonnet|Bonnet]]
(1843), [[Biography:Carl Johan Malmsten|Malmsten]] (1846, 1847, the latter without integration); Stokes (1847), Paucker (1852), Chebyshev (1852), and Arndt
(1853).

General criteria began with [[Biography:Ernst Kummer|Kummer]] (1835), and have been studied by [[Biography:Gotthold Eisenstein|Eisenstein]] (1847), Weierstrass in his various
contributions to the theory of functions, [[Biography: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|uniform convergence]] was treated by Cauchy (1821), his limitations being pointed out by Abel, but the first to attack it
successfully were Seidel and 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 [[Biography: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>

Genocchi (1852) has further contributed to the theory.

Among the early writers was Wronski, whose "loi suprême" (1815) was hardly recognized until [[Biography: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
[[Biography:Jacob Bernoulli|Jacob Bernoulli]] (1702) and his brother [[Biography:Johann Bernoulli|Johann Bernoulli]] (1701) and still
earlier by Vieta. Euler and Lagrange simplified the subject,
as did Poinsot, [[Biography:Karl Schröter|Schröter]], [[Biography:James Whitbread Lee Glaisher|Glaisher]], and [[Biography:Ernst Kummer|Kummer]].

Fourier (1807) set for himself a different problem, to
expand a given function of ''x'' in terms of the sines or cosines of
multiples of ''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. [[Biography: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 Cauchy (1826) to
attempt and for Dirichlet (1829) to handle in a thoroughly
scientific manner (see [[Convergence of Fourier series|convergence of Fourier series]]). Dirichlet's treatment (''Crelle'', 1829), of trigonometric series was the subject of criticism and improvement by
Riemann (1854), Heine, [[Biography:Rudolf Lipschitz|Lipschitz]], [[Biography:Ludwig Schläfli|Schläfli]], and
du Bois-Reymond. Among other prominent contributors to the theory of
trigonometric and Fourier series were [[Biography:Ulisse Dini|Dini]], [[Biography:Charles Hermite|Hermite]], [[Biography:Georges Henri Halphen|Halphen]],
Krause, Byerly and [[Biography:Paul Émile Appell|Appell]].

==Generalizations==

=== Asymptotic series ===

Asymptotic series, otherwise [[Asymptotic expansion|asymptotic expansion]]s, are infinite series whose partial sums become good approximations in the limit of some point of the domain.  In general they do not converge, but they are useful as sequences of approximations, each of which provides a value close to the desired answer for a finite number of terms. The difference is that an asymptotic series cannot be made to produce an answer as exact as desired, the way that convergent series can. In fact, after a certain number of terms, a typical asymptotic series reaches its best approximation; if more terms are included, most such series will produce worse answers.

=== Divergent series ===

{{Main|Divergent series}}
Under many circumstances, it is desirable to assign a limit to a series which fails to converge in the usual sense.  A summability method is such an assignment of a limit to a subset of the set of divergent series which properly extends the classical notion of convergence.  Summability methods include [[Cesàro summation]], (''C'',''k'') summation, Abel summation, and [[Borel summation]], in increasing order of generality (and hence applicable to increasingly divergent series).

A variety of general results concerning possible summability methods are known.  The [[Silverman–Toeplitz theorem]] characterizes ''matrix summability methods'', which are methods for summing a divergent series by applying an infinite matrix to the vector of coefficients.  The most general method for summing a divergent series is non-constructive, and concerns [[Banach limit]]s.

=== Summations over arbitrary index sets ===

Definitions may be given for sums over an arbitrary index set <math>I.</math><ref>{{citation|author=Jean Dieudonné|title=Foundations of mathematical analysis|publisher=Academic Press}}</ref>  There are two main differences with the usual notion of series: first, there is no specific order given on the set <math>I</math>; second, this set <math>I</math> may be uncountable.  The notion of convergence needs to be strengthened, because the concept of [[Conditional convergence|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|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|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, define

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

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|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>

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.

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

==== 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|topological group]] <math>X.</math> 
Let <math>\operatorname{Finite}(I)</math> be the collection of all [[Finite set|finite]] [[Subset|subset]]s of <math>I,</math> with <math>\operatorname{Finite}(I)</math> viewed as a [[Directed set|directed set]], [[Partially ordered set|ordered]] under inclusion <math>\,\subseteq\,</math> with [[Union (set theory)|union]] as join. 
The family <math>\left(a_i\right)_{i \in I},</math> is said to be {{em|unconditionally summable}} if the following limit, which is denoted by <math>\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 \left\{\sum_{i\in A} a_i \,: A \subseteq I, A \text{ finite }\right\}</math>
Saying that the sum <math>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|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|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, 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|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>\sum a_n</math> is unconditionally summable, then the series remains convergent after any [[Permutation|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>\sum a_n</math> converges, then the series is unconditionally convergent.  When <math>X</math> is 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|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'''{{sfn|Schaefer|Wolff|1999|pp=179–180}} if the limit <math>\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|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|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>\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>\left(\sum_{n=0}^N x_n\right)_{N=1}^{\infty}</math> converges to <math>x</math> in <math>X</math>; to wit,

<math display=block>\left\|x - \sum_{n=0}^N x_n\right\| \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|topological group]]. 
Specifically, in this case, <math>\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|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> <p>a function whose [[Support (mathematics)|support]] is a [[Singleton (mathematics)|singleton]] <math>\{a\}.</math>  Then</p> <math display=block>f = \sum_{a \in X}f_a</math> <p>in the topology of pointwise convergence (that is, the sum is taken in the infinite product group <math>Y^X</math>).</p>
# 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> <p>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.</p>
# On the [[First uncountable ordinal|first uncountable ordinal]] <math>\omega_1</math> viewed as a topological space in the [[Order topology|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> <p>(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.</p>

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

==References==
{{Reflist}}

==Bibliography==
{{Refbegin}}
*Bromwich, T. J. ''An Introduction to the Theory of Infinite Series'' MacMillan & Co. 1908, revised 1926, reprinted 1939, 1942, 1949, 1955, 1959, 1965.
* {{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
}}
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn | Narici | 2011 | p=}} -->
* {{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}}
* Walter Rudin, ''Principles of Mathematical Analysis'' (McGraw-Hill: New York, 1964).
* {{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 }} <!-- {{sfn | Pietsch | 1972 | p=}} -->
* {{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 | oclc=589250 }} <!-- {{sfn | Robertson | 1973 | p=}} -->
* {{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 }} <!-- {{sfn | Ryan | 2002 | p=}} -->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn | Schaefer | 1999 | p=}} -->
* {{Trèves François Topological vector spaces, distributions and kernels}} <!-- {{sfn | Treves | 2006 | p=}} -->
* {{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 }} <!-- {{sfn | Wong | 1979 | p=}} -->
{{MathSciNet|id=0033975}}
{{Refend}}

==External links==
* {{springer|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)}}
{{Analysis-footer}}


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

{{Sourceattribution|Series (mathematics)}}