Is $f(x) = \sum_{n=2}^{+\infty} \frac{\cos(nx)}{n^2\log(n)}$ a $C^1$ function?

Question

Let $f$ be defined on the real line by $$f(x) = \sum_{n=2}^{+\infty} \frac{\cos(nx)}{n^2\log(n)}$$ Is $f$ a $C^1$ function?

I manage to prove that it is $C^1$ on $\mathbb R \setminus 2\pi \mathbb Z$ using Abel's summation technique.

I manage to prove that it is differentiable in $0$ (or at multiples of 2$\pi$) by showing that $\frac{f(x) - f(0)}{x} \to_{x \to 0} 0$.

However, what about the continuity of $f'$ at point $0$?

Answer 1

Hint: your series is uniformly converged. For derivative series works Dirichlet's test

Answer 2

The given series and the series of derivative $-\sum\frac {\sin (nx)} {n \log n}$ are both uniformly convergent. For uniform convergence of the later see the section '(C) and (S) series' in Fourier Series by Edwards. (Theorem 7.2.2 Partb (1), p. 112). The uniform convergence of these two series implies that your function does belong to $C^{1}$.