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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 $\mathrm{If\:}\lim _{n\to \infty }a_n\ne 0\mathrm{\:then\:}\sum a_n\mathrm{\:diverges}$

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$\frac{2n^2}{5n^2+2n+1}$ 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?

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That’s exactly right. Well done.

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