Cauchy’s root test

If $\sum a_n$ is a series of positive real terms and

for all $n>N$, then $\sum a_n$ is convergent. If $\sqrt[n]{a_n}\ge 1$ for an infinite number of values of $n$, then $\sum a_n$ is divergent.

Limit form

Given a series $\sum a_n$ of complex terms, set

$$\rho =\underset{n\to \mathrm{\infty }}{lim\; sup}\sqrt[n]{|a_n|}$$

The series $\sum a_n$ is absolutely convergent if and is divergent if $\rho >1$. If $\rho =1$, then the test is inconclusive.