# 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:

• root test (Cauchy’s root test),

• ratio test of d’Alembert (http://planetmath.org/RatioTestOfDAlembert),

• ratio test,

• $p$-test (http://planetmath.org/PTest),

• Raabe’s criteria.

• 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.