Editing Series (mathematics) (section)

===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]].