Proving the convergence/divergence of $\sum_{n=1}^\infty \frac{\cos\ n}{n} (1+\frac{1}{\sqrt{2}}+...+\frac{1}{\sqrt{n}})$ Do the following series converges? Why?
$$ \sum_{n=1}^ \infty  \frac{\cos\ n}{n} (1+\frac{1}{\sqrt{2}}+\cdots+\frac{1}{\sqrt{n}})$$

Hint. One may apply Dirichlet's test. 
One may prove that,
$$
n \mapsto \frac1{n} \left(\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+...+\frac{1}{\sqrt{n}}\right)
$$ is decreasing to $0$. Then use the fact that
$$
\left|\sum_{k=1}^n \cos k\right|=\left|\Re \sum_{k=1}^n e^{ik}\right|=\left|\Re \frac{e^{i}\left(e^{in}-1\right)}{e^{i}-1}\right|\leq \frac2{1-\cos(1)}.
$$
