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Thread: Simplify this infinite convergent sum.

  1. #1
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    Simplify this infinite convergent sum.

    .


    Simplify the infinite convergent sum as much as possible in terms of \phi:

    \dfrac{1}{\phi^1} + \dfrac{1}{\phi^2} + \dfrac{1}{\phi^3} \  + \  ...


    where \phi equals  \dfrac{\sqrt{5} + 1}{2}.





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    Source:

    The Call Of The Primes: Surprising Patterns, Peculiar Puzzles, and Other Marvels of Mathematics

    by Owen O'Shea

    pages 123, 125-126
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  2. #2
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    Re: Simplify this infinite convergent sum.

    This is a geometric series:

    $\sum_{n\ge 1}\left(\dfrac{1}{\phi}\right)^n = \left[\sum_{n\ge 0}\left(\dfrac{1}{\phi}\right)^n\right]-1=\dfrac{1}{1-\dfrac{1}{\phi}}-1 = \dfrac{1}{\phi-1}=\phi $
    Last edited by SlipEternal; Jun 5th 2017 at 09:51 AM.
    Thanks from greg1313
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