Convergence or not of infinite series: $\sum^{\infty}_{n=1}\frac{n}{1+n^2}$ How can we prove that the series $\displaystyle \sum^{\infty}_{n=1}\frac{n}{1+n^2}$ is convergent or divergent?
Solution I try:
$$\lim_{m\rightarrow \infty}\sum^{m}_{n=1}\frac{n}{1+n^2}<\lim_{m\rightarrow \infty}\sum^{m}_{n=1}\frac{n}{n^2}$$
Did not know how I can solve that problem from that point.

Hint : For $n>1:$
$\dfrac{n}{1+n^2} \gt \dfrac{n}{n^2+n^2} = (1/2)\dfrac{1}{n}.$
Hence?
