Convergence tests

In mathematics, convergence tests are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence or divergence of an infinite series.

List of tests

Limit of the summand

If the limit of the summand is undefined or nonzero, that is , then the series must diverge. In this sense, the partial sums are Cauchy only if this limit exists and is equal to zero. The test is inconclusive if the limit of the summand is zero.

Ratio test

This is also known as D'Alembert's criterion. Suppose that there exists such that

Root test

This is also known as the nth root test or Cauchy's criterion. Define r as follows:

Integral test

The series can be compared to an integral to establish convergence or divergence. Let be a non-negative and monotone decreasing function such that . If

Direct comparison test

If the series is an absolutely convergent series and for sufficiently large n , then the series converges absolutely.

Limit comparison test

If , and the limit exists, is finite and is not zero, then converges if and only if converges.