Linear Combination of Convergent Series
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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 $\ds\sum_{n \mathop = 1}^\infty a_n$ and $\ds \sum_{n \mathop = 1}^\infty b_n$ converge to $\alpha$ and $\beta$ respectively.
Let $\lambda, \mu \in \R$ be real numbers.
Then the series $\ds \sum_{n \mathop = 1}^\infty \paren {\lambda a_n + \mu b_n}$ converges to $\lambda \alpha + \mu \beta$.
Proof
\(\ds \sum_{n \mathop = 1}^N \paren {\lambda a_n + \mu b_n}\) | \(=\) | \(\ds \lambda \sum_{n \mathop = 1}^N a_n + \mu \sum_{n \mathop = 1}^N b_n\) | Linear Combination of Indexed Summations | |||||||||||
\(\ds \) | \(\to\) | \(\ds \lambda \alpha + \mu \beta \text{ as } N \to \infty\) | Combination Theorem for Sequences |
$\blacksquare$
Sources
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 6.8$