determining series convergence

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

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

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 http://planetmath.org/?op=getobj&from=lec&id=37this article.

Title	determining series convergence
Canonical name	DeterminingSeriesConvergence
Date of creation	2013-03-22 13:24:45
Last modified on	2013-03-22 13:24:45
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	16
Author	CWoo (3771)
Entry type	Topic
Classification	msc 40A05
Related topic	ThenA_kto0IfSum_k1inftyA_kConverges
Related topic	LimitComparisonTest
Related topic	AbsoluteConvergence
Related topic	InfiniteProductOfSums1a_i