Sum of Absolutely Convergent Series

Theorem
Let $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ and $\displaystyle \sum_{n \mathop = 1}^\infty b_n$ be two real or complex series that are absolutely convergent.

Then the series $\displaystyle \sum_{n \mathop = 1}^\infty \paren {a_n + b_n}$ is absolutely convergent, and:


 * $\displaystyle \sum_{n \mathop = 1}^\infty \paren {a_n + b_n} = \displaystyle \sum_{n \mathop = 1}^\infty a_n + \sum_{n \mathop = 1}^\infty b_n$

Proof
Let $\epsilon \in \R_{>0}$.

From Tail of Convergent Series tends to Zero, it follows that there exists $M \in \N$ such that:
 * $\displaystyle \sum_{n \mathop = M + 1}^\infty \cmod {a_n} < \dfrac \epsilon 2$

and:
 * $\displaystyle \sum_{n \mathop = M + 1}^\infty \cmod {b_n} < \dfrac \epsilon 2$

For all $m \ge M$, it follows that:

By definition of convergent series, it follows that:

To show that $\displaystyle \sum_{n \mathop = 1}^\infty \paren {a_n + b_n}$ is absolutely convergent, note that: