Manipulation of Absolutely Convergent Series

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Theorem

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


Permutation

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

$\displaystyle \sum_{n \mathop = 1}^\infty a_n = \sum_{n \mathop = 1}^\infty a_{\pi \left({n}\right)}$


Characteristic Function

Let $A \subseteq \N$.

Then:

$\displaystyle \sum_{n \mathop = 1}^\infty a_n \chi_A \left({n}\right) = \sum_{n \mathop \in A} a_n$

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


Scale Factor

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

Then:

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