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

Ratio test

Let \( \sum_{n=m}^{\infty}a_n \) be a series of non-zero numbers.

• If [...], then the series \( \sum_{n=m}^{\infty}a_n \) is absolutely convergent (hence conditionally convergent).
• If [...], then the series \( \sum_{n=m}^{\infty}a_n \) is not conditionally convergent (hence not absolutely convergent).
• Otherwise, we cannot assert any conclusion.

The non-zero requirement is needed to avoid a division by zero.