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Zeta function and summation convergence

  1. Jun 27, 2009 #1
    I need to know if the following series converges:

    ∑(k=1 to k=oo)[(((-1)^k) ζ(k))/(e^k)]

    The problem is that zeta(1)=oo; however, the equation satisfies the conditions of convergence for an alternating series [the limit as k->oo=0 and each term is smaller than the last.]

    Any thoughts?
    Last edited: Jun 27, 2009
  2. jcsd
  3. Jun 27, 2009 #2
    How did you arrive at the sum?
  4. Jun 27, 2009 #3
    well yeah I see what you're saying about zeta of 1. To see if the summation converges, try one of the tests, like tha ratio test.
  5. Jun 28, 2009 #4
    This one converges
    \sum_{k=2}^\infty \frac{(-1)^k \zeta(k)}{e^k}

    But in the original zeries, the [tex]k=1[/tex] term is the problem.
  6. Jun 28, 2009 #5
    Yes if memory serves me right that sum is just a constant and an x away from being a taylor series of the digamma function.
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