determining series convergence
Consider a series $\mathrm{\Sigma}{a}_{n}$. To determine whether $\mathrm{\Sigma}{a}_{n}$ 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 $\mathrm{\Sigma}{a}_{n}$ 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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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 $\mathrm{\Sigma}{a}_{n}$ 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  20130322 13:24:45 
Last modified on  20130322 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 