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.
•
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.

单词	DeterminingSeriesConvergence
释义	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. • When the terms in $\\Sigma a_{n}$ are positive, there are several possibilities: – comparison test, – root test (Cauchy’s root test), – ratio test of d’Alembert (http://planetmath.org/RatioTestOfDAlembert), – ratio test, – $p$ -test (http://planetmath.org/PTest), – integral test, – Raabe’s criteria. • limit comparison test. • the divergence test (http://planetmath.org/ThenA_kto0IfSum_k1inftyA_kConverges). • If the series is an alternating series, then the alternating series test (http://planetmath.org/AlternatingSeriesTest) may be used. • 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.
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