Ratio test

Let $\sum _{n=m}^{\mathrm{\infty }}{a}_n$ be a series of non-zero numbers.

• If [...], then the series $\sum _{n=m}^{\mathrm{\infty }}{a}_n$ is absolutely convergent (hence conditionally convergent).
• If [...], then the series $\sum _{n=m}^{\mathrm{\infty }}{a}_n$ 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.