Prove $ \sum \frac{\cos n} { \sqrt n}$ converges

Question

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 ?

Answer 1

Use Dirichlet's test. If $\{a_n\}$ is a sequence whose partial sums are bounded, and $\{b_n\}$ is a decreasing sequence with $\lim b_n=0$, then $\sum a_nb_n$ converges.

Answer 2

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.