real analysis - If ${a_{n}}>0$ and $sumlimits_{n=1}^{infty}a_n$ diverge

If $\{a_{n}\}>0$ and $\sum\limits_{n=1}^{\infty}a_n$ diverge.

The following series: converge, diverge, or neither?

$\sum\limits_{n=1}^{\infty} \dfrac{a_n}{1 + a_{n^2}} , \sum\limits_{n=1}^{\infty} \dfrac{a_n}{1 + na_{n}}$ and $\sum\limits_{n=1}^{\infty} \dfrac{a_n}{a_{n} +n^2 a-{n}}$ ?

1) $\sum\limits_{n=1}^{\infty} \dfrac{a_n}{1 + na_{n}}$

let $a_{n} = \frac{1}{n}.$ then $\frac{a_n}{1 + na_{n}} = \frac{1}{2n}$

$\sum\limits_{n=1}^{\infty} \dfrac{a_n}{1 + na_{n}} =$ $\sum\limits_{n=1}^{\infty} \frac{1}{2n}$ diverge .

Is this reasoning correct?