Convergent Series can be Added Term by Term
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Theorem
Let $\ds \sum_{n \mathop = 1}^\infty a_n$ and $\ds \sum_{n \mathop = 1}^\infty b_n$ be two convergent series converging to $A$ and $B$ respectively.
Then:
- $\ds \sum_{n \mathop = 1}^\infty \paren {a_n + b_n} = A + B$
Proof
\(\ds A + B\) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty a_n + \sum_{n \mathop = 1}^\infty b_n\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{N \mathop \to \infty} \sum_{n \mathop = 1}^N + \lim_{N \mathop \to \infty} \sum_{n \mathop = 1}^N b_n\) | Definition of Convergent Series of Numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{N \mathop \to \infty} \sum_{n \mathop = 1}^N a_n + \sum_{n \mathop = 1}^N b_n\) | Sum Rule for Real Sequences | |||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{N \mathop \to \infty} \sum_{n \mathop = 1}^N \paren {a_n + b_n}\) | rearranging a finite sum | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \paren {a_n + b_n}\) | Definition of Convergent Series of Numbers |
$\blacksquare$