Cauchy’s root test

If $\sum a_{n}$ is a series of positive real terms and

\sqrt[n]{a_{n}}<k<1

for all $n>N$ , then $\sum a_{n}$ is convergent. If $\sqrt[n]{a_{n}}\geq 1$ 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=\limsup_{n\to\infty}\sqrt[n]{|a_{n}|}

The series $\sum a_{n}$ is absolutely convergent if $\rho<1$ and is divergent if $\rho>1$ . If $\rho=1$ , then the test is inconclusive.