1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook