Math and science::Analysis::Tao::07. Series

# The Root Test

Let $$\sum_{n=m}^{\infty}a_n$$ be a series of real numbers and let [ $$\alpha = ?$$ ].

1. If $$\alpha < 1$$, then the series $$\sum_{n=m}^{\infty}a_n$$ is absolutely convergent (and hence conditionally convergent).
2. If $$\alpha > 1$$, then the series $$\sum_{n=m}^{\infty}a_n$$ 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.