Does the sum of the series $\sum\limits_{n=1}^\infty\frac{\sin nx^2}{1+n^3}$ define a continuously differentiable function on $\mathbb R ?$

Question

$f(x)=\displaystyle\sum_{n=1}^\infty\frac{\sin nx^2}{1+n^3}$

MY TRY:We know that $\sum_{n=1}^{\infty}\frac{\sin nx^2}{1+n^3} \le \sum_{n=1}^{\infty}\frac{1}{n^3+1}$.So the function is uniformly convergent.

Now derivative of this function is $\sum_{n=1}^\infty \frac{2nx\cos (nx^2)}{1+n^3}.$But what can we conclude from this$?$Thank you.

Answer 1

Hint: Show that $\sum_{n=1}^\infty \frac{2nx\cos (nx^2)}{1+n^3}$ is locally uniformly convergent and hence the initial series define a continuously differentiable function on $\mathbb {R}$. QED