Let be $P$ and $Q$ two polynomials, and $deg Q =deg P +1$ (so, if $P=kx^2+...$, then $Q= jx^3+...$).
Let be $R=P/Q$.
Show that for every $\alpha>0$, there is a correct choice of $n(\alpha)$ such that every term of the sum $$S_{\alpha}=\sum_{n=n(\alpha)}^{\infty} (-1)^n R\left(2+n^{\alpha} \right)$$ is defined. And show when this sum converges.