Proving the function $F(x)$ is differentiable

Question

Kindly asking how to prove that $F(x)=\sum_{n=1}^\infty \sin(nx)/\sinh(n\pi)$ is differentiable for all $x$ and of any order? I am really stcuk to treat this function. Please make me a right path. Thank you.

Answer 1

Differentiating the series term by term leads to a reasonable guess that $$F'(x)=\sum_{n=1}^\infty\frac{n\cos(nx)}{\sinh(n\pi)}.$$ Notice that the $n$th term is no greater than $n/\sinh(n\pi)$, and the sum of these bounds converges. So the series converges uniformly. Now, I hope the fact that uniformly convergent series can be integrated termwise is part of your knowledge base! Use the fundamental theorem of calculus to finish up.