In the past few years I have become interested on a type of questions I have stumbled upon. Here are some posts that have produced various results: Problem 1 Basic example with exponential, Problem 2 Basic example with polinoial,Problem 3 General example,Problem 4 Prime Numbers series,Problem 5 with denominator multpiplied by a loghartihm. All these are series where the denominator is a polinomial (times a logharithm on problem 5) and the numerator contains a certain function to which a modulus operation is applied to respect to another polinomial of a degree smaller by one in comparison to the denominator. Let us focus a bit on this series: $$\sum_ {n = 1}^\infty \frac {(f(n) +P(n)) \pmod{ Q(n)}} {D(n)}$$, which is the third problem discussed above. This series heuristically diverges. I have tried now more restrictive conditions to try to break the heuristic. I'd like to discuss the following series: $$\sum_ {n = 1}^\infty \frac {(-1)^k(f(n) +P(n)) \pmod{ Q(n)}} {D(n)}$$, where $$k=n \pmod 3$$ and $$deg(Q(n))=deg(D(n))-1$$. I think this series diverges as there are twice positive terms than negative terms. That dispite 1 positive and 1negative terms rougly cancelling out, there are probably enough terms to diverge. Am I correct in arguing like this for a weaker but still correct heuristic for the divergence on the title series?
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