Show that the series of functions $\sum\limits_{n\geq 1} {x\over n(1+nx^2)} $ is uniformly convergent for all real $x$

Question

Show that the series of functions $\sum\limits_{n\geq 1} {x\over n(1+nx^2)} $ is uniformly convergent for all real $x$.

My work.
I find out that at $ x=\dfrac{1}{\sqrt n}$, $ {\dfrac{x} {n(1+nx^2)}}$ is maximum and from there I am able to show the uniform convergence of the given series of function. But I am not able to find the sequence $\ \{M_n\}_n$ to perform Weierstrass M-test.

Answer 1

Let $\ f_n (x)$ = $x\over n(1+nx^2)$. Then I found that $f_n (x)$ is maximum at $x= \frac {1}{\sqrt{n}}$. Thus $$|f_n (x)| \leq \frac {\frac{1}{\sqrt{n}}}{2n} = \frac {1}{2n^{3/2}}.$$ Now $\sum_{n\geq 1} \frac {1}{2n^{3/2}}$ is convergent for all real $x$, by p-test, as $\frac{3}{2} \gt 1$. So the given series is uniformly convergent for all real $\ x$.