Manipulation of Absolutely Convergent Series/Scale Factor

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Theorem

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


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

Then:

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


Proof

\(\displaystyle c \sum_{n \mathop = 1}^\infty a_n\) \(=\) \(\displaystyle c \lim_{N \mathop \to \infty} \sum_{n \mathop = 1}^N a_n\)
\(\displaystyle \) \(=\) \(\displaystyle \lim_{N \mathop \to \infty} \sum_{n \mathop = 1}^N c a_n\) Multiple Rule for Sequences
\(\displaystyle \) \(=\) \(\displaystyle \sum_{n \mathop = 1}^\infty c a_n\)

$\blacksquare$