Ratio Test - Knowlpedia

Ratio test: Let $\sum_{n=1}^\infty a_n$ be a series with $a_n\ne0$ for all sufficiently large $n$ , and define

L^+=\limsup_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|, \qquad L^-=\liminf_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|.

If $L^+<1$ , then $\sum a_n$ is absolutely convergent.
If $L^->1$ , then $a_n\not\to0$ , so $\sum a_n$ diverges.
In all other cases, these bounds alone are inconclusive.

In particular, if the ratio has a limit $L$ , the series converges absolutely for $L<1$ , diverges for $L>1$ , and the test is inconclusive for $L=1$ .

Remarks

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.