determining series convergenceDeterminingSeriesConvergence2003-02-01 21:01:192008-04-19 14:46:17Topic% this is the default PlanetMath preamble. as your knowledge
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\def\ser{\Sigma a_n}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 the comparison test,
\item the root test (Cauchy's root test),
\item the ratio test,
\item \PMlinkname{$p$-test}{PTest},
\item the integral test.
\item Raabe's criteria.
\end{itemize}
\item The 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}