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Thread: Prove P(x) >= M

  1. #1
    Oct 2009

    Prove P(x) >= M

    Let P be a polynomial with positive leading coefficient:
    P(x) = a_nx^n +a_{n-1}x^{n-1} + \ldots + a_1x + a_0, n>=1
    Clearly, as x\to\infty,\ a_nx^n\to\infty. Show that, as x --> infinity, P(x)\to\infty by showing that, given any positive number M, there exists a positive number K such that if x>=K, then P(x) >=M.

    (Hint: you might wish to choose  K_i>0 such that
    |a_{n-i}/x^i|< a_n/2n when x > K_i and then take the largest of the K_i. )
    Last edited by Opalg; Nov 22nd 2009 at 01:00 AM. Reason: rescued LaTeX
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