Combination Rule for Series

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Theorem

Let the two series $\displaystyle \sum_{n=1}^\infty a_n$ and $\displaystyle \sum_{n=1}^\infty b_n$ converge to $\alpha$ and $\beta$ respectively.

Let $\lambda, \mu \in \R$ be real numbers.

Then the series $\displaystyle \sum_{n=1}^\infty \left({\lambda a_n + \mu b_n}\right)$ converges to $\lambda \alpha + \mu \beta$.

	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \sum_{n=1}^N \left({\lambda a_n + \mu b_n}\right)$	$=$	$\displaystyle \lambda \sum_{n=1}^N a_n + \mu \sum_{n=1}^N b_n$	$\displaystyle $	$\displaystyle $	$\displaystyle $
	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\to$	$\displaystyle \lambda \alpha + \mu \beta \text{ as } N \to \infty$	$\displaystyle $	$\displaystyle $	$\displaystyle $	from Combination Theorem for Sequences

$\blacksquare$

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \sum_{n=1}^N \left({\lambda a_n + \mu b_n}\right)\)	\(=\)	\(\displaystyle \lambda \sum_{n=1}^N a_n + \mu \sum_{n=1}^N b_n\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\to\)	\(\displaystyle \lambda \alpha + \mu \beta \text{ as } N \to \infty\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	from Combination Theorem for Sequences