I know that we can show that $$\sum\limits_{n=1}^{\infty} \frac{\cos n} {\sqrt n}$$ converges using Dirichlet's test.
However, how come the following shows the divergence of the same series? Where is the error?
Note that $-1 \leq \cos n \leq 1$. Hence,
$$\sum\limits_{n=1}^{\infty}\frac{\cos n}{\sqrt n}\geq\sum\limits_{n=1}^{\infty}\frac{-1}{\sqrt n}$$
But we know that $\sum \frac{-1} {\sqrt n} $
diverges as it is a $p$-series. Hence $\sum \frac{\cos n} {\sqrt n}$ must also diverge by the comparison test.
Best Answer
Be careful with applying the comparison test, the terms of these series don't have a fixed sign.
If $0 \le a_n \le b_n$ and $\sum a_n$ is divergent, then so is $\sum b_n$; in your case, not all the criteria are met.