Does the series $f'(x) = \sum\limits_{n = 1}^{\infty} -\frac{1}{n} e^{-nx} \cos(e^{-nx})$ converge uniformly or pointwise on $\mathbb{R}$?

Question

I would say no, since for $x < 0$, $f'(x) \rightarrow \infty$ as $n \rightarrow \infty$ since $e^{-nx} \rightarrow \infty$ as $n \rightarrow \infty$. So the series doesn't even converge for $x < 0$.

Also $f'(0) = \sum\limits_{n=1}^{\infty} -\frac{1}{n} \cos(1)$ which would seem to diverge to $-\infty$.

Correct?