Does the derivative of the Fourier series $\sum_{k\geq2}\frac{1}{k^2\log k}\cos kt$ exist?
I wanted to use the following sufficient condition for this problem
Theorem. Let $a_k(t)$ be differentiable. If a series of functions $\sum_{k}a_k(t)$ is such that $\sum_ka_k'(t)$ converges uniformly in $[a,b]$, and $\sum_{k}a_k(t)$ converges at ont point in $(a,b)$, then $\sum_ka(t)$ converges in $[a,b]$ and its derivative is $\sum_ka_k'(t)$.
But this theorem does not seem to be applicable to the series above, because I can't prove that $-\sum_k\frac{1}{k\log k}\sin kt$ is uniformly convergent. However, a book that I am reading says that it does have a derivative. How can I prove it?