Tell if a series converges uniformly

Question

Let $f(x) = \sum_{n=1}^\infty \frac{-2x}{(x^2+n^2)^2}$. Check if $f_n(x)$ converges to a continuous function.

So I've seen a solution that uses the fact that if $f(x)$ converges uniformly and $f_n(x)$ are continuous, so does $f(x)$.

The author search for maximum points of $\frac{-2x}{(x^2+n^2)^2}$. Afterward, he applied the two maximum points, $x_{M_1}, x_{M_2}$ for $f(x)$, and showed that $f(x)$ converges at those points.

He reached the conclusion that $f(x)$ converges uniformly.

What's the theorem behind that?

Answer 1

The author used the Weierstrass M-test theorem.