Proving the convergence/divergence of $\sum_{n=1}^\infty \frac{\cos\ n}{n} (1+\frac{1}{\sqrt{2}}+...+\frac{1}{\sqrt{n}})$

Question

Do the following series converges? Why?

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

Answer 1

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 $$ \left|\sum_{k=1}^n \cos k\right|=\left|\Re \sum_{k=1}^n e^{ik}\right|=\left|\Re \frac{e^{i}\left(e^{in}-1\right)}{e^{i}-1}\right|\leq \frac2{1-\cos(1)}. $$

Answer 2

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,

$nS_{n+1}-(n+1)S_n=nS_{n} + \frac{n}{\sqrt{n+1}} - (n+1)S_n = \frac{n}{\sqrt{n+1}} - S_n<0$.