I want to show that the following function is continuously differentiable:
$g(x)=\sum_{n=1}^{\infty}\frac{1}{n^{2}}e^{\int_{0}^{x}t\sin(n/t)\,dt}.$
I tried using the idea that series where the first n terms of the infinite series converge uniformly if it converges pointwise at a time and derivative series converges uniformly. So I tried to show derivative series converges by the Cauchy criterion but I can't seem to bound the tail.