TITLE: Infinite series with cos in numerator
QUESTION [5 upvotes]: How do you evaluate this series?
$$\sum_{i=1}^{\infty}\frac{\cos i}{2^i}$$
It's absolutely convergent by comparison to the geometric series. But the $\cos$ is tripping me up. I've tried differentiating in order to go through the $\cos$ -> $\sin$ -> $\cos$ route, but that gives me different powers of 2 in the denominator. Any ideas?

REPLY [5 votes]: $$ S=\sum_{n\geq 1}\frac{\cos n}{2^n}=\text{Re}\sum_{n\geq 1}\left(\frac{e^i}{2}\right)^n = \text{Re}\left(\frac{e^i}{2-e^i}\right)=\text{Re}\left(\frac{2e^i-1}{5-4\cos 1}\right)=\color{red}{\frac{2\cos 1-1}{5-4\cos 1}}. $$