Summations and Infinite Series, oh my!

Algebra Level 3

$(\sum\limits_{n=1}^\infty \frac{5 ^ n}{16 ^ n})+\frac{(\frac{4}{3}+\frac{4}{9}+\frac{4}{27}+\frac{4}{81}+...)}{44}$ can be written as $\frac{a}{b}$ , where $a$ and $b$ are coprime positive integers. Find the value of $\sum\limits_{n=0}^4 a^n+b^n$ .