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Closed form for (in)finite sums

  1. Apr 17, 2013 #1
    I have a set of questions concerning the perennial sum
    [tex]\large \sum_{k=1}^{n}k^p[/tex]
    and its properties.

    1. For [itex]p \ge 0[/itex], the closed form of this is known (via Faulhaber's formula).
    I know little about divergent series, but I've read that in some sense there exists a value associated with these sums as [itex]n \rightarrow \infty[/itex]: for instance [itex]\lim_{n \rightarrow \infty}\sum_{k=1}^{n}k = \frac{-1}{12}[/itex] (here)? I think this is called zeta function regularisation.

    a. What is the significance of such sums? Are they useful realistic compatable-with-rest-of-mathematics, or wholly artificial? What evidence is there of these pseudosums' existence?

    b. What are they for [itex]p \ge -1[/itex] (hoping that the Harmonic series, [itex]p = -1[/itex], has such a 'pseudo-sum')?

    2. Moving on to [itex]p < 0[/itex]. For even [itex]p[/itex] the limit as [itex]n \rightarrow \infty[/itex] infinity is known (Euler).

    a. Is there a Faulhaber equivalent in this case (a closed form of the sum up to [itex]n[/itex] for either even or integer [itex]n[/itex])?

    b. Are the odd [itex]p [/itex]'s infinite series likely to be found?

    For comparison, with constant [itex]k [/itex] and changing [itex]p [/itex]:
    [tex]\large \sum_{p=1}^{n}k^p= \frac{1-k^{p+1}}{1-k}[/tex]
    I would love if for constant [itex]p [/itex] and changing [itex]k [/itex] was a formula as complete as this.

    Thank you for volunteering to suffer my question.
    Last edited: Apr 17, 2013
  2. jcsd
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