The Root Test

Let $\sum _{n=m}^{\mathrm{\infty }}{a}_n$ be a series of real numbers and let [ $\alpha =?$ ].

1. If $\alpha <1$, then the series $\sum _{n=m}^{\mathrm{\infty }}{a}_n$ is absolutely convergent (and hence conditionally convergent).
2. If $\alpha >1$, then the series $\sum _{n=m}^{\mathrm{\infty }}{a}_n$ is not conditionally convergent (and hence is not absolutely convergent either).
3. If $\alpha =1$, this test does not assert any conclusion.

The famous Root Test.