Uniform convergence of series $\sum_{n=1}^{\infty}\frac{x}{n(1+nx^2)}$ 
Discuss the uniform convergence of the series $$\sum_{n=1}^{\infty}\frac{x}{n(1+nx^2)}$$ for real values of $x$.

Let $f(x)$ denote the given series. Then $f(0)=0+0+\ldots = 0$, and $f(-x)=-f(x)$ for all real $x$. Also, $$f(1) = \sum_{n=1}^{\infty}\frac{1}{n(1+n)} =  \sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{1}{n+1}\right) = 1.$$ For any $x$ we can write $$\frac{x}{n(1+nx^2)} = \frac{x}{n}-\frac{x^3}{1+nx^2},$$ so $$f(x) = \sum_{n=1}^{\infty}\left(\frac{x}{n}-\frac{x^3}{1+nx^2}\right).$$ How can we check which values of $x$ this sum converges?

Hint: $$n(1+nx^2) \ge 2n \sqrt{n}|x|.$$
