Manipulation of Absolutely Convergent Series
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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:
- $\ds \sum_{n \mathop = 1}^\infty a_n = \sum_{n \mathop = 1}^\infty a_{\map \pi n}$
Characteristic Function
Let $A \subseteq \N$.
Then:
- $\ds \sum_{n \mathop = 1}^\infty a_n \map {\chi_A} n = \sum_{n \mathop \in A} a_n$
where $\chi_A$ is the characteristic function of $A$.
Scale Factor
Let $c \in \R$, or $c \in \C$.
Then:
- $\ds c \sum_{n \mathop = 1}^\infty a_n = \sum_{n \mathop = 1}^\infty c a_n$