Absolute convergence, and the absolute convergence test 

Absolute convergence, definition

Let $\sum _{n=m}^{\mathrm{\infty }}{a}_n$ be a formal series of real numbers. The series is said to be absolutely convergent if the series $\sum _{n=m}^{\mathrm{\infty }}|a_n|$ is convergent.

In order to distinguish convergence from absolute convergence, we sometimes refer to the former as conditional convergence.

Absolute convergence test

Let $\sum _{n=m}^{\mathrm{\infty }}{a}_n$ be a formal series of real numbers. If the series is absolutely convergent, then it is also conditionally convergent. Furthermore, in this case we have the triangle inequality

Note that the converse of this proposition is not true; there exists series that are conditionally convergent but are not absolutely convergent.