Ratio test

In mathematics, the ratio test is a criterion for convergence or divergence of a series whose terms are real or complex numbers. It considers the ratio of successive terms of the series; if the ratio tends to a limit less than 1 for terms further along the series, then it converges. Formally, it states that if

then the series

converges, and if

then the series diverges. In particular, if

exists, then the series converges if that limit is < 1 and diverges if it is > 1. There exist both convergent and divergent series for which the limit is exactly 1, and consequently the test is inconclusive in that case.

Example

Consider the series:

1 + 1/2 + 1/4 + 1/8 ...

The first ratio to consider is the second term divided by the first, which is simply 1/2. The second ratio is the third term divided by the second, which is 1/4 divided by 1/2, which is 1/2. Since in fact each term is half of its precedor, the ratio will always be 1/2. Thus the limit of these ratios is 1/2, and the series converges.