# Is this convergent or divergent: $\sum _{n=1}^{\infty }\:\frac{2n^2}{5n^2+2n+1}$?

Is this convergent or divergent? if convergent, find the sum. If divergent explain why.

$$\sum _{n=1}^{\infty }\:\frac{2n^2}{5n^2+2n+1}$$

I want to use the divergent test which is $\mathrm{If\:}\lim _{n\to \infty }a_n\ne 0\mathrm{\:then\:}\sum a_n\mathrm{\:diverges}$

$\frac{2n^2}{5n^2+2n+1}$ I would assume to take out the largest $n$ in both the top and bottom

$\lim _{n\to \infty }\left(\frac{n^2}{n^2}\cdot \frac{2}{5+\frac{2}{n}+\frac{1}{n^2}}\right)\:$

having the $n^2$ cancel

$\lim _{n\to \infty }\frac{2}{5+\frac{2}{n}+\frac{1}{n^2}}$ after taking the limit

$\frac{2}{5+0+0}$

so $\frac{2}{5}$$\ne$ $\:0\:$ so it’s divergent?