Consider $$f_C(x)=\sum_{n=2}^\infty \frac{\cos(2\pi nx)}{n\log n}$$ Show that it does not converges uniformly and $f_C(x)\geq c\log \log \frac{1}{|x|}$ as $x\rightarrow 0$.
I used summation parts formula: let $a_n=\frac{1}{n\log n}$ and $b_n=\cos(2\pi n x)$, I got the series to be non uniformly convergent: as $x\rightarrow 0$ \begin{align*} |S_M-S_N|&=\left |a_Mb_M-a_Nb_N+\sum_{n=N}^{M-1}B_n(a_{n+1}-a_n)\right|\\ &=\left|\frac{1}{M\log M}-\frac{1}{N\log N}+\sum_{n=N}^{M-1}\frac{1}{(n+1)\log(n+1)}-\frac{1}{n\log n}\right|\\ &=2|\frac{1}{M\log M}-\frac{1}{N\log N}|\\ &=2(\frac{1}{N\log N}-\frac{1}{M\log M})\\ \end{align*} So the series is uniformly divergent but how could you show the inequality? Thanks