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

• When the terms in $\ser$ are positive, there are several possibilities:
  • the comparison test,
  • the root test (Cauchy's root test),
  • the ratio test,
  • $p$ -test,
  • the integral test.
  • Raabe's criteria.
• The limit comparison test.
• The divergence test.
• If the series is an alternating series, then the alternating series test may be used.
• Abel's test for convergence can be used when terms in $\ser$ 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 this article.