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Growth rate of integer power sum

  1. Jan 29, 2010 #1
    I need to show that

    [tex]\sum_{i=0}^n i^k=\Theta(n^{k+1})[/tex]

    Or equivalently

    [tex] \lim_{n\to\infty}\frac{\sum_{i=0}^n i^k}{n^{k+1}}=C[/tex]

    I simply don't know what to do with the sum here. I know that I can rewrite or expand it, but that doesn't seem to help me. Any suggestions?

    Thank you!
  2. jcsd
  3. Jan 29, 2010 #2


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    You could think of the integer sum as a lower sum for an integral of f(x)=x^k.
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