# Convergence of the series $\sum_ {n\geq1} \frac {(f(n) +P(n)) \pmod {Q(n)}} {D(n)}$

Under what conditions does the series $$\sum_ {n = 1}^\infty \frac {(f(n) +P(n)) \pmod{ Q(n)}} {D(n)}$$ converge?

Here $$\text{P}$$, $$\text{Q}$$ and $$\text{D}$$ are polynomials, with $$\text{deg}(Q)= \text{deg}(D)-1$$.

Also $$f(n)= \displaystyle \sum_{r=2}^{k} a_{r} r^n$$ where $$\{a_{r} \}_{r=2}^k$$ are real numbers and k is a natural number with k>2. k and the degree of P must be larger than the degree of Q.

Here and here particular cases of this question have been answered by the user @SangchulLee. I'd like to know in which cases this series converges. (I assume they are rare but Sangchul Lee has already found some). I'm getting somewhere with this so don't flag me yet!