Let $a_{n}$ and $b_{n}$ be sequences of real numbers for $n \geqslant 1$ such that $\left|a_{n}\right| \leqslant c / n^{1+\epsilon}$ and $\left|b_{n}\right| \leqslant c / n^{1+\epsilon}$ for all $n \geqslant 1$, for some constants $c>0$ and $\epsilon>0$. Show that the series

$$f(x)=\sum_{n \geqslant 1} a_{n} \cos n x+\sum_{n \geqslant 1} b_{n} \sin n x$$

converges uniformly to a continuous function on the real line. Show that $f$ is periodic in the sense that $f(x+2 \pi)=f(x)$.

Now suppose that $\left|a_{n}\right| \leqslant c / n^{2+\epsilon}$ and $\left|b_{n}\right| \leqslant c / n^{2+\epsilon}$ for all $n \geqslant 1$, for some constants $c>0$ and $\epsilon>0$. Show that $f$ is differentiable on the real line, with derivative

$$f^{\prime}(x)=-\sum_{n \geqslant 1} n a_{n} \sin n x+\sum_{n \geqslant 1} n b_{n} \cos n x .$$

[You may assume the convergence of standard series.]