sequences and series - How do I evaluate this sum: :$sum_{n=3}^{infty} frac{1}{1-n+n^2-cdots+(-n)^k}$ and $k$ is positive integer?

Wolfram alpha showed after some calculations for evaluation of this series :

$\sum_{n=3}^{\infty} \frac{1}{1-n+n^2-\cdots+(-n)^k}$ for example for $k=10$ ,

I have got this result which it close to $0$.

My question here is : : How do I evaluate this sum: :$$\sum_{n=3}^{\infty} \frac{1}{1-n+n^2-\cdots+(-n)^k}$$ ?

Note: I exclude the singularity points just i would like to Know how do i evaluate it

Thank you for any help