Manipulation of Absolutely Convergent Series/Scale Factor

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

\(\ds \)

\(=\)

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

$\blacksquare$