Results 1 to 1 of 1

Thread: [SOLVED] Rearrangements of alternating harmonic series

  1. #1
    Newbie
    Joined
    Nov 2009
    Posts
    7

    [SOLVED] Rearrangements of alternating harmonic series

    Hey guys, I'm pretty stuck on the following problem. It involves rearrangements of the terms in the alternating harmonic series to produce different sums:

    Let s = 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + ...

    a) Show that s/2 = 1 - 1/2 - 1/4 + 1/3 - 1/6 - 1/8 + 1/5 - 1/10 - 1/12 + ...
    b) Show that 3s/2 = 1 + 1/3 - 1/2 + 1/5 + 1/7 - 1/4 + 1/9 + 1/11 - 1/6 + ...

    I think I've got part (a) so far. If you look at each group of three terms, pairing the first two terms in that group together and computing the difference seems to do the trick.
    (1 - 1/2) - 1/4 + (1/3 - 1/6) - 1/8 + (1/5 - 1/10) - 1/12 + ...
    = 1/2 - 1/4 + 1/6 - 1/8 + 1/10 - 1/12 + ...
    = 1/2 (1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + ...)
    = s/2

    I'm stuck on part (b) though. I've tried many different groupings, and also tried using the result from part (a) and adding it to the given information to try and get the result, but I just can't seem to get it. Any tips or advice on this part would be greatly appreciated. Thanks!

    EDIT: Finally figured it out. I subtracted the sum for s/2 from the sum for 3s/2, term-by-term, and showed that those terms were in the right order to produce the sum s. I thought I tried this earlier but I guess I just got hung up and confused on the grouping aspect of it.
    Last edited by tonyc4l; Jan 25th 2010 at 10:03 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Clarification on harmonic series
    Posted in the Calculus Forum
    Replies: 3
    Last Post: Feb 17th 2010, 06:21 PM
  2. Alternating series
    Posted in the Calculus Forum
    Replies: 4
    Last Post: Jan 31st 2010, 10:44 PM
  3. Alternating Harmonic Series
    Posted in the Calculus Forum
    Replies: 6
    Last Post: Dec 16th 2009, 07:21 AM
  4. Replies: 3
    Last Post: Dec 12th 2009, 07:31 AM
  5. Replies: 2
    Last Post: Nov 18th 2009, 11:06 PM

Search Tags


/mathhelpforum @mathhelpforum