Manipulation of Absolutely Convergent Series/Scale Factor
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Theorem
Let $\ds \sum_{n \mathop = 1}^\infty a_n$ be a real or complex series that is absolutely convergent.
Let $c \in \R$, or $c \in \C$.
Then:
- $\ds c \sum_{n \mathop = 1}^\infty a_n = \sum_{n \mathop = 1}^\infty c a_n$
Proof
\(\ds c \sum_{n \mathop = 1}^\infty a_n\) | \(=\) | \(\ds c \lim_{N \mathop \to \infty} \sum_{n \mathop = 1}^N a_n\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{N \mathop \to \infty} \sum_{n \mathop = 1}^N c a_n\) | Multiple Rule for Sequences | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty c a_n\) |
$\blacksquare$