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

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.

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 $$\ x$$.