continuity of function series

Question

So here it is:

$$\sum_{n=1}^\infty \frac{\sin(\frac{1}{nx^2})}{1+(x-1)\ln^4(xn)}$$

$$x \in (1,\infty)$$

My task is to prove its continuity if possible.
My lead was to try proving it through uniform convergence, what do you think, is it possible?

Answer 1

Hint: It's enough to show the series converges uniformly on any $[a,\infty),$ where $a>1.$ To do this, try the Weierstrass M test. I'm seeing the $n$th term as $\le \sin(1/n)/[(a-1)(\ln n)^4]$ for $n>1$ on $[a,\infty).$