Root test

Root test: For a series $\sum_{n=1}^\infty a_n$, define

where $|a_n|$ denotes the absolute value of the term.

• If $L<1$, then $\sum_{n=1}^\infty a_n$ is absolutely convergent (hence convergent ).
• If $L>1$ (including $L=\infty$), then $\sum_{n=1}^\infty a_n$ is divergent .
• If $L=1$, the test is inconclusive.

The root test is especially natural for power series and is closely related to the Cauchy–Hadamard theorem ; compare also the ratio test .