# Math Help - more summation help

1. ## more summation help

Let P and Q be polynomials of degree p and q. Suppose that the coefficient of x^p in P is positive, the coefficient of x^q in Q is positive, and that Q(k) does not equal zero for all positive integers k. Prove that the series summation from k=1 to infinity of P(K)/Q(k) converges if and only if p < q-1.

Thanks for any help!

2. Originally Posted by friday616
Let P and Q be polynomials of degree p and q. Suppose that the coefficient of x^p in P is positive, the coefficient of x^q in Q is positive, and that Q(k) does not equal zero for all positive integers k. Prove that the series summation from k=1 to infinity of P(K)/Q(k) converges if and only if p < q-1.

Thanks for any help!
$\frac{P(x)}{Q(x)}\sim\frac{1}{x^{q-p}}$. Apply firstly the limit comparison test (after making a neccessary remark about eventual the eventual sign of $\frac{P(x)}{Q(x)}$), and secondly the ratio, integral, root, prety much anything test.