Editing Series (mathematics) (section)

===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.