Manipulation of Absolutely Convergent Series

Theorem
If a sum is absolutely convergent, all of the following are true:


 * $\displaystyle \sum_{n=1}^\infty a_n = \sum_{n \in \N} a_n$, where in the second expression the terms are taken in any order


 * $\displaystyle \sum_{n=1}^\infty a_n \chi_A = \sum_{n \in A} a_n$, where $A$ is a set and $\chi_A$ its characteristic function


 * $\displaystyle b \sum a_n = \sum b a_n$