Math and science::Analysis::Tao::07. Series
Absolute convergence, and the absolute convergence test
Absolute convergence, definition
Let \( \sum_{n=m}^{\infty}a_n \) be a formal series of real numbers. The series is said to be absolutely convergent if the series \( \sum_{n=m}^{\infty}|a_n| \) is convergent.
In order to distinguish convergence from absolute convergence, we sometimes refer to the former as conditional convergence.
Absolute convergence test
Let \( \sum_{n=m}^{\infty}a_n \) 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
\[ \left| \sum_{n=m}^{\infty}a_n \right| \le \sum_{n=m}^{\infty}|a_n| \]
Note that the converse of this proposition is not true; there exists series that are conditionally convergent but are not absolutely convergent.
