infinite series containing cos(x/n)

Question

for $n \in \{ 1,2,3,\dots \}$ we have $\sum\frac{(-1)^{n+1}\cos(\frac{x}{n})}{n}$. If integrated term by term twice, the resulting series clearly diverges. Does this mean the original series diverges as well? This is not a strictly alternating signs series and those have been difficult to prove the convergence/divergence of them.

Answer 1

For any specific $x$, it is eventually alternating. Also,$$\frac d{dn}\frac{\cos(x/n)}{n}$$ is eventually negative, so the alternating series test applies, and it converges.