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?
$$ \sum_{n=1}^ \infty  \frac{\cos\ n}{n} (1+\frac{1}{\sqrt{2}}+\cdots+\frac{1}{\sqrt{n}})$$

Let $\frac{1}{1}+\cdots + \frac{1}{\sqrt{n}}=S_n$. If we can show that the sequence $\frac{S_n}{n}$ is decreasing towards $0$ then we are done, by the Dirichlet's test, since $M_n = \sum_{i=1}^n \cos n$ is bounded. We can proceed directly,
$nS_{n+1}-(n+1)S_n=nS_{n} + \frac{n}{\sqrt{n+1}} - (n+1)S_n = \frac{n}{\sqrt{n+1}} - S_n<0$.
