Find out whether the series $\sum_{n=1}^{\infty} \left(e^{\frac{\cos n}{n^{3/8}}}-1\right)$ converges

Question

There is series $$\sum_{n=1}^{\infty} \left(e^{\frac{\cos n}{n^{3/8}}}-1\right)$$

I expanded it into Taylor series: $$\sum_{n=1}^{\infty}{\frac{\cos n}{n^{3/8}}+\frac{\cos^2 n}{n^{3/4}} +r_n}$$ Where $r_n$ is remainder.

Remainder converges, the task is to prove that first or second fraction converges or not.

What should I do now?

Answer 1

HINT:

Note that

$$\frac{\cos^2(n)}{n^{3/4}}=\frac{1+\cos(2n)}{2n^{3/4}}$$

and the series $\sum_{n\ge 1}n^{-3/4}$ diverges.