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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.

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

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  • $\begingroup$ Thank you for your answer, after reading your answer, I know how to prove the convergence. $\endgroup$
    – UESTCfresh
    Mar 21 at 9:45

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