Summations and Infinite Series, oh my!

Algebra Level 3

(n=15n16n)+(43+49+427+481+...)44(\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 ab\frac{a}{b}, where aa and bb are coprime positive integers. Find the value of n=04an+bn\sum\limits_{n=0}^4 a^n+b^n.

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