Ratio Test

Given an infinite series \(S\)
\[ S = \sum_{i=0}^\infty s_i \]
for which the limit of the absolute value of the ratio of \(s_{n+1}\) to \(s_n\) converges to a value \(r\) as \(n\) tends to infinity
\[ r = \lim_{n \to \infty} \left|\frac{s_{n+1}}{s_n}\right| \]
then if \(r\) is less than one the series converges and if it is greater than one it diverges. If it equals one then no conclusion can be drawn from the test.