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.

• When the terms in $\mathrm{\Sigma }{a}_n$ 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 $\mathrm{\Sigma }{a}_n$ 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.