# Manipulation of Absolutely Convergent Series

## Theorem

Let $\ds \sum_{n \mathop = 1}^\infty a_n$ be a real or complex series that is absolutely convergent.

### Permutation

If $\pi: \N \to \N$ is a permutation of $N$, then:

$\displaystyle \sum_{n \mathop = 1}^\infty a_n = \sum_{n \mathop = 1}^\infty a_{\map \pi n}$

### Characteristic Function

Let $A \subseteq \N$.

Then:

$\displaystyle \sum_{n \mathop = 1}^\infty a_n \map {\chi_A} n = \sum_{n \mathop \in A} a_n$

where $\chi_A$ is the characteristic function of $A$.

### Scale Factor

Let $c \in \R$, or $c \in \C$.

Then:

$\ds c \sum_{n \mathop = 1}^\infty a_n = \sum_{n \mathop = 1}^\infty c a_n$