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Simple serie Q

  1. Nov 29, 2004 #1


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    Hi. I am starting the study of series and I don't see how to do this problem.

    "Show that

    [tex]\sum_{n=0}^\infty \frac{1}{(n+a)(n+1+a)} = \frac{1}{a}[/tex]"

    All i got is the decomposition in partial fractions as

    [tex]\sum_{n=0}^\infty (...) = \sum_{n=0}^\infty \frac{1}{(n+a)} + \sum_{n=0}^\infty \frac{-1}{(n+1+a)}[/tex]

    if these sum converge. I tried seeing a patern in the partial sums to find [itex]S_n[/itex] but it's too difficult so there must be another way.

    Any hint/help will be appreciated.
    Last edited: Nov 29, 2004
  2. jcsd
  3. Nov 29, 2004 #2
    Using your partial fractions decomposition and rearranging the terms gives:

    [tex]\frac{1}{a} - \frac{1}{a+1} + \frac{1}{a+1} - \frac{1}{a+2} + \ldots[/tex]

    It can't be any more obvious now can it.
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