Ratio Test

A series converges absolutely if successive terms shrink by a uniform factor less than one.

Ratio Test

Ratio test: Let $\sum_{n=1}^\infty a_n$ be a series with $a_n\ne 0$ for all sufficiently large $n$ , and define

L=\limsup_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|.

If $L<1$ , then $\sum a_n$ is absolutely convergent (hence convergent).
If $L>1$ (including $L=\infty$ ), then $\sum a_n$ is divergent .
If $L=1$ , the test is inconclusive.

The ratio test is particularly effective for factorials, exponentials, and power-series-like terms, and it is closely related to the root test and the Cauchy–Hadamard theorem for power series .