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Math Help - more summation help

  1. #1
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    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.

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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by friday616 View Post
    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.
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