Nature of the series $\sum_{n \geq 2} \frac{1}{1+(3^n+P(n))\operatorname{mod}(n^2)}$, where $P(n)$ is a $3$rd degree polynomial

Asked Dec 27 '23 at 13:27

Active Dec 27 '23 at 13:52

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A long time ago I had asked this: question

and then the generalization discussed: here

Now I'm wondering about the situation where the mod operator is at the denominator and not the numerator. This seems even more challenging to me.

The heuristic discussed in the second link above does not help much. It may be said that $(3^n+P(n))\operatorname{mod}(n^2)$ is evenly distributed but this does not help at all because the formula I just mentioned could yield an $0$ or a small enough number that makes the fraction large enough to make the series divergent provided this happens often enough. The likelihood of a large element of the series to appear decreases but as the number of "attempts" is infinite it will still happen as far as I can see. So the question is: Is this intuition correct and the title series is divergent or there is more to it?

edited Dec 27 '23 at 13:52

asked Dec 27 '23 at 13:27

Aurelian Florea

Nature of the series $\sum_{n \geq 2} \frac{1}{1+(3^n+P(n))\operatorname{mod}(n^2)}$, where $P(n)$ is a $3$rd degree polynomial

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