Series of reprocicals of polynomials

Question

Find which polynomials $p$ with real coefficients and no natural roots verify the following: $\sum_{i=1}^{\infty} \frac{1}{p(i)}$ converges. Is there any result related to this? I don't want a complete solution, just a hint.

Can somebody help me with this, please?

I found a document somewhat related, but it hasn't helped me much:

@lulu Obviously natural integers. If $p$ has such roots, the series is not defined. — Jean-Claude Arbaut, Dec 10 '17 at 19:58
No natural number is a root of $p$, otherwise we would get 1 divided by zero in our infinite sum. — Asix, Dec 10 '17 at 19:59
Hint: as $n\to\infty$, $p(n)$ is equivalent to the term of highest degree, say $a_mn^m$. — Jean-Claude Arbaut, Dec 10 '17 at 20:00
I think the question is quite close to this question, and the series $\sum_{n=1}^{\infty}\frac{1}{n^{1+\epsilon}}$ converges for all $\epsilon>0$. — Dietrich Burde, Dec 10 '17 at 20:06
I think the answer is all of them having degree greater than 1 — WW1, Dec 10 '17 at 20:18
@Jean-ClaudeArbaut So, since the dominant coefficient only gives us the sign of the limit, we can assume $p$ is a monomial. Taking your hint into account, can I say that the limit of that series is equal to the limit of the p-harmonic series? And from there, by the comparison test with series of reprocicals of squares, we get that $p$ must be of degree greater than 1? Is that the only condition for $p$ that we can get? — Asix, Dec 10 '17 at 20:39
Yes. If $u_n\sim v_n$, then $\sum_nu_n$ converges iff $\sum_nv_n$ converges. Hence you can keep only the monomial, etc. And yes, this is the only condition on $p$. — Jean-Claude Arbaut, Dec 11 '17 at 14:22

score 2 · Accepted Answer · answered Dec 10 '17 at 21:06

All linear or constant polynomials will diverge. The linear ones will diverge logarithmically. All polynomials of higher degree will converge.

The important lesson from this is that convergence or divergence only depends on what happens "near infinity". As $i$ gets large the non-leading terms of the polynomial matter less and less, so they do not contribute to the question of convergence. They will change the limit, but not whether the whole thing converges or not. We know that the sum of $\frac 1i$ diverges and that the sum of $\frac 1{i^2}$ converges, which is all we need for the first paragraph.

Series of reprocicals of polynomials

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