\( \newcommand{\cat}[1] {\mathrm{#1}} \newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})} \newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}} \)

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.