Editing Series (mathematics) (section)

==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>