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?
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