TITLE: How to prove $\sum_{n=1}^\infty{(\sqrt{n}-2)^2 \over n^2+\sqrt{n^4+1} }$ diverges?
QUESTION [0 upvotes]: $\sum_{n=1}^\infty{(\sqrt{n}-2)^2 \over n^2+\sqrt{n^4+1} }$
I tried using the ratio test but i cant cancel anything out.

REPLY [2 votes]: By equivalence:
$$(\sqrt n-2)^2\sim_\infty\sqrt n^2=n,\enspace n^2+\sqrt{n^4+1}\sim_\infty 2n^2, \enspace\text{whence}\quad\frac{(\sqrt n-2)^2}{n^2+\sqrt{n^4+1}}\sim_\infty \frac{n}{2n^2}=\frac1{2n},$$
which diverges.
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