# Difference 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:

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

## Proof

The series $\displaystyle \sum_{n \mathop = 1}^\infty \paren {-b_n}$ is absolutely convergent, as $\cmod {-b_n} = \cmod {b_n}$ for all $n \in \N$.

Then:

 $\displaystyle \sum_{n \mathop = 1}^\infty a_n - \sum_{n \mathop = 1}^\infty b_n$ $=$ $\displaystyle \sum_{n \mathop = 1}^\infty a_n + \paren {-1} \sum_{n \mathop = 1}^\infty b_n$ $\displaystyle$ $=$ $\displaystyle \sum_{n \mathop = 1}^\infty a_n + \sum_{n \mathop = 1}^\infty \paren {-1} b_n$ Manipulation of Absolutely Convergent Series: Scale Factor $\displaystyle$ $=$ $\displaystyle \sum_{n \mathop = 1}^\infty \paren {a_n - b_n}$ Sum of Absolutely Convergent Series

$\blacksquare$