Let $f: \mathbb R \to \mathbb R, f(x)= \sum^{\infty}_{k=1}\frac{(-1)^{k}\cos(kx)}{k^{2}}$,
show that $f$ is differentiable in $]-\pi,\pi[$ using the following hint: $x = \sum^{\infty}_{k=1}(-1)^{k}\frac{2}{k}\sin(kx)$ $ \forall x \in ]-\pi, \pi[$
I have used the differential quotient so far but no luck, any hints?
Furthermore, I am then asked to calculate $\sum^{\infty}_{k=1}\frac{1}{k^{2}}$ using the above.