TITLE: Prove $ \sum \frac{\cos n} { \sqrt n}$ converges
QUESTION [5 upvotes]: How to Prove $ \sum \frac{\cos n} { \sqrt n}$ converges Using Abel’s theorem ?
I think it can be done using $\cos n = Re(e^{in})$ { Real Part of Complex Number }
How to proceed ?

REPLY [4 votes]: Notice that $\sum\limits_{k=1}^{n} \cos k = \frac{\sin(n+\frac{1}{2})-\sin\frac{1}{2}}{2\sin\frac{1}{2}}$. Hence the sum is bounded. And since $\frac{1}{\sqrt{n}}$ is decreasing to zero, then the series converge.