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Convergent series

  1. Nov 12, 2009 #1
    1. The problem statement, all variables and given/known data

    For what values of p does the series [1/1^p - 1/2^p + 1/3^p - 1/4^p +... converge?

    2. Relevant equations



    3. The attempt at a solution

    I believe that this series converges for all p \in N because the sequence of a_n's is nonincreasing and converges to 0. I am not quite sure, however, to show that it converges to 0. I know that the sequence 1/n converges to 0 and I know that p is fixed, but I don't know how to massage that information into what I need. Thanks.
     
  2. jcsd
  3. Nov 12, 2009 #2
    If the series is alternating, then you only need to show that [tex] |a_n| \leq |a_{n-1}| [/tex] and that the sequence of ans converges to zero.

    A sequence converges to zero if for any positive real number [tex] \epsilon [/tex], you can find a N large enough so that [tex] a_n < \epsilon [/tex] for all n>N.
     
  4. Nov 12, 2009 #3
    Right; I understand the epsilon proof and the theorem related to alternating series. Although I know it sounds really dumb, I am having trouble finding N.
     
  5. Nov 12, 2009 #4
    [tex] \frac{1}{N^p} = \epsilon. [/tex]

    Now, for certain kinds of p you can always find an N for every epsilon.
     
  6. Nov 12, 2009 #5
    I was thinking for p >= 0. Thank you so much.
     
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