Determine whether a function series is uniformly convergent

Question

Determine whether $\sum_{j=0}^{\infty} \frac{\sin(jx)}{(2+x^2)^j}$ is uniformly convergent for $x\in\mathbb{R}$

So I started by saying as $|\sin(jx)|\le1$ so $\sum_{j=0}^{\infty} \frac{\sin(jx)}{(2+x^2)^j} \leq \sum_{j=0}^{\infty} \frac{1}{(2+x^2)^j} = \sum_{j=0}^{\infty}(\frac{1}{(2+x^2)})^j=\sum_{j=0}^{\infty}\frac{1}{1-\frac{1}{2+x^2}}=\frac{2+x^2}{1+x^2} $ So the function is pointwise convergent to the limit function $f(x)=\frac{2+x^2}{1+x^2}$

How do I now prove whether or not it is uniformly convergent? Thanks

Answer 1

Hint: $$\bigg|\frac{\sin(jx)}{(2+x^2)^j}\bigg|\leq \frac{1}{2^j}$$ now use Weierstrass M-test.