40A05-DeterminingSeriesConvergence.tex

\documentclass[12pt]{article}
\usepackage{pmmeta}
\pmcanonicalname{DeterminingSeriesConvergence}
\pmcreated{2013-03-22 13:24:45}
\pmmodified{2013-03-22 13:24:45}
\pmowner{CWoo}{3771}
\pmmodifier{CWoo}{3771}
\pmtitle{determining series convergence}
\pmrecord{16}{33958}
\pmprivacy{1}
\pmauthor{CWoo}{3771}
\pmtype{Topic}
\pmcomment{trigger rebuild}
\pmclassification{msc}{40A05}
\pmrelated{ThenA_kto0IfSum_k1inftyA_kConverges}
\pmrelated{LimitComparisonTest}
\pmrelated{AbsoluteConvergence}
\pmrelated{InfiniteProductOfSums1a_i}

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\def\ser{\Sigma a_n}
\begin{document}
Consider a series $\ser$. To determine whether $\ser$ converges or diverges, several tests are available. There is no precise rule indicating which \PMlinkescapetext{type} of test to use with a given series. The more obvious approaches are collected below.

\begin{itemize}
\item When the terms in $\ser$ are positive, there are several possibilities:
\begin{itemize}
\item comparison test,
\item root test (Cauchy's root test),
\item \PMlinkname{ratio test of d'Alembert}{RatioTestOfDAlembert},
\item ratio test,
\item \PMlinkname{$p$-test}{PTest},
\item integral test,
\item Raabe's criteria.
\end{itemize} 
\item limit comparison test.
\item \PMlinkname{the divergence test}{ThenA_kto0IfSum_k1inftyA_kConverges}.
\item If the series is an alternating series, then the \PMlinkname{alternating series test}{AlternatingSeriesTest} may be used.
\item Abel's test for convergence can be used when terms in $\ser$ can be obained as the product of terms of a convergent series with terms of a monotonic convergent sequence.
\end{itemize}

The root test and the ratio test are direct applications of the comparison test to the geometric series with terms $(|a_n|)^{1/n}$ and $\frac{a_{n+1}}{a_n}$, respectively.

For a paper about tests for convergence, please see \PMlinkexternal{this article.}{http://planetmath.org/?op=getobj&from=lec&id=37}
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