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Thread: finite alternating harmonic series

  1. #1
    MHF Contributor
    Dec 2012
    Athens, OH, USA

    finite alternating harmonic series

    This is similar to an existing thread, but sufficiently different, I think.
    For any positive integer n, let S(n)=\Sigma^n_{k=1}\left({1\over2k-1}-{1\over2k}\right)={u_n\over d_n} with u_n and d_n relatively prime. The question concerns u_n and d_n.

    One result. The exact power of 2 that divides d_n is floor(log2(n))+1.

    Here's the proof:
    finite alternating harmonic series-mhfalternatingharmonic.png

    What else can one say about the sequence d_n? The exact power of 3 that divides d_n seems to be an increasing sequence, but I can't find any closed formula as for 2. The exact power of 5 that divides d_n is not even an increasing sequence.

    The integers u_n and d_n get quite large. I know almost nothing about the u_n.
    u_2, u_3, u_5, u_8 and u_9 are prime.

    Java has a pretty sophisticated probalistic prime testing algorithm. Java says with probability greater than 1-one millionth that u_{254} is prime; u_{254} has 221 decimal digits.

    (I inadvertenly attached a thumbnail and don't know how to delete it; ignore it.)
    Attached Thumbnails Attached Thumbnails finite alternating harmonic series-alternatingseries.png  
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