Linear Combination of Convergent Series

Theorem

Let $\sequence {a_n}_{n \mathop \ge 1}$ and $\sequence {b_n}_{n \mathop \ge 1}$ be sequences of real numbers.

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

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

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

Proof

 $\displaystyle \sum_{n \mathop = 1}^N \paren {\lambda a_n + \mu b_n}$ $=$ $\displaystyle \lambda \sum_{n \mathop = 1}^N a_n + \mu \sum_{n \mathop = 1}^N b_n$ $\quad$ Linear Combination of Indexed Summations $\quad$ $\displaystyle$ $\to$ $\displaystyle \lambda \alpha + \mu \beta \text{ as } N \to \infty$ $\quad$ Combination Theorem for Sequences $\quad$

$\blacksquare$