Difference of Absolutely Convergent Series

Theorem

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

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

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

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

Then:

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

$=$

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

$\ds $

$=$

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

$\ds $

$=$

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

$\blacksquare$