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 \(\sum\limits_{n=0}^4 a^n+b^n\).

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