Let $P$ and $Q$ be polynomials of degree $k$ and $m$ and suppose $Q(n)\ne0\forall n\in\mathbb N$.
Prove that $\sum_{n=1}^\infty\frac{P(n)}{Q(n)}$ is convergent if $m\ge k+2$ and divergent if $m\le k+1$.
Can someone point me in the right direction? I have tried to transform into something telescoping and see if the terms cancel each other out but with no result.
Other than that, the only thing I can think of is trying induction but I feel as if there most likely is a much better way to prove it using other thereoms. Possibly the tail lemma.
If $m \ge k+2$ and $n$ is large enough, then there exists $c_1>0$ such that
$$\left| \frac{Q(n)}{P(n)}\right| \ge c_1n^{m-k} \ge c_1n^2$$
that is $$\left| \frac{P(n)}{Q(n)}\right| \le \frac{c_1}{n^{2}}$$
Hence by comparison test, we have the result.
Also, if $m \le k+1$, WLOG, we assume that $P$ and $Q$ have positive leading coefficient. (Suppose not, we study its negative instead).
If $n$ is large enough, there exists $c_2>0$ such that $$\frac{P(n)}{Q(n)} \ge c_2n^{k-m} \ge \frac{c_2}{n}$$
Again, by comparison test, we have the result.