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Math Help - Prove series result?

  1. #1
    Super Member fardeen_gen's Avatar
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    Prove series result?

    If S_p = 1^p + 2^p + 2^p + \mbox{...} + n^p, prove that S_0 + (p + 1)S_1 + \frac{p^2 + p}{2}S_2 + \mbox{...} + (p + 1)S_p = (n + 1)^{p + 1} - 1
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  2. #2
    Senior Member TheAbstractionist's Avatar
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    Quote Originally Posted by fardeen_gen View Post
    If S_p = 1^p + 2^p + \color{red}3\color{black}^p + \mbox{...} + n^p, prove that S_0 + (p + 1)S_1 + \frac{p^2 + p}{2}S_2 + \mbox{...} + (p + 1)S_p = (n + 1)^{p + 1} - 1
    Hi fardeen_gen.

    By the binomial theorem,

    (k+1)^{p+1}-k^{p+1}\ =\ 1+(p+1)k+\frac{(p+1)p}2k^2+\cdots+(p+1)k^p

    Now sum both sides from k=1 to n and you have your answer.
    Last edited by TheAbstractionist; May 22nd 2009 at 12:17 PM.
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