# How to prove that this series which is summation of rational polynomial fractions is absolutely convergent?

Consider the sequence $$a_n= \frac{n^3+2n^2+2n+4}{n^5+n^4+7n^2+1}$$,$$n \geq0$$.

Prove that the series $$\sum_{n \geq0}{a_n}$$ is absolutely convergent.

I try to prove it using ratio test, but when I try to calculate the limit $$\lim_{n \to \infty}|\frac{a_{n+1}}{a_n}|$$, I find that it equals 1, and I can't use the ratio test.

Can someone give me some hints?

Now I try to use compare test, however, I have no idea about finding the right series to compare to. I really want to know if there are some principles for finding such series. I think that exercises of limits and series are too tricky for me, they are too flexible, many times when I meet such exercise I have no direction to work in.

Note, that $$|a_n|\leq c/n^2$$ for all $$n$$ where $$c$$ is suitable large. This may be seen by writing $$a_n=\frac{1+2/n+2/n^2+4/n^3}{n^2(1+1/n+7/n^3+1/n^5)}$$.