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

$\blacksquare$