$ \sum_{n=1}^{\infty}a_{n} $ diverges but $ \sum_{n=1}^{\infty}\frac{a_{n}}{1+a_{n}^{2}} $ sometimes converges and sometime diverges.

Let $a_n = 2^n$. Then $\sum_{n = 1}^\infty a_n$ diverges. However, since $a_n/(1 + a_n^2) < 1/2^n$ and $\sum_{n = 1}^\infty 1/2^n$ converges, by the comparison test, $\sum_{n = 1}^\infty a_n/(1 + a_n^2)$ converges.

Hint. Let $r>1$. If $a_n>r^n$ then $\sum_{n=1}^{\infty}a_n=\infty$. Note that $$ \sum_{n=1}^{\infty} \frac{a_n}{1+a_n^2} = \sum_{n=1}^{\infty} \frac{1}{\frac{1}{a_n}+a_n} \leq \sum_{n=1}^{\infty}\frac{1}{r^n} = \lim_{n\to \infty}\frac{(\frac{1}{r})^{n+1}-\frac{1}{r}}{(\frac{1}{r})-1} = \frac{-\frac{1}{r}}{\frac{1}{r}-1} = \frac{1}{r-1} $$