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Divergent series question

  1. Dec 23, 2005 #1
    let be the divergent series:

    [tex]1^p+2^p+3^p+.....................+N^p=S(N) [/tex] with p>0 my

    question is..how i would prove that this series S would diverge in the form:

    [tex]S(N)=N^{p+1}/p+1 [/tex] N--->oo

    for the cases P=1,2,3,... i can use their exact sum to prove it but for the general case i can not find any prove..perhaps i should try Euler sum formula ..are the divergent series S(N) equal to the integral:

    [tex]\int_{0}^{\infty}dxx^{p} [/tex] they both diverge in the same way.
    Last edited: Dec 23, 2005
  2. jcsd
  3. Dec 23, 2005 #2


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    Homework Helper

    Try approximating the integral:

    [tex]\int_0^1 x^p dx [/tex]

    with strips of width 1/N.
  4. Dec 28, 2005 #3


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    You still seem to be having trouble distinguishing between an equality and an asymptotic.

    You are apparently trying to show that what you've called S(N) is asymptotic to [tex]N^{p+1}/(p+1)[/tex]? Just compare S(N) with the integrals

    [tex]\int_0^N x^p dx[/tex] and [tex]\int_1^{N+1} x^p dx[/tex].

    Euler-Maclaurin summation will work as well, but is not needed for this asymptotic.
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