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Math Help - Geometric Series Problem

  1. #1
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    Geometric Series Problem

    Hey guys, I am currently in a DiffEq class, and I can't remember how to do Geometric Series for the life of me. I appreciate any help you can give me.

    ∑[(e^(-(i+1)πs)+e^(-iπs))/(s^2+1)]

    The summation is from 0 to infinity.
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  2. #2
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    Quote Originally Posted by eg37se View Post
    Hey guys, I am currently in a DiffEq class, and I can't remember how to do Geometric Series for the life of me. I appreciate any help you can give me.

    ∑[(e^(-(i+1)πs)+e^(-iπs))/(s^2+1)]

    The summation is from 0 to infinity.
    What's the index n or s?
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  3. #3
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    Ohhh yea sorry, it looks confusing.

    ∑[(e^(-(i+1)πs)+e^(-iπs))/(s^2+1)]


    s is a constant
    i is the index
    π=pi
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  4. #4
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    Your series is \frac{1}{1+s^2} \sum_{i=0}^\infty e^{-(i+1) \pi s} + e^{- i \pi s} which can be written as

     <br />
\frac{1 + e^{\pi s}}{1+ s^2} \sum_{i=0}^\infty e^{- i \pi s} = <br />
\frac{1 + e^{\pi s}}{1+ s^2} \sum_{i=0}^\infty \left(e^{- \pi s} \right)^i<br />
= <br />
\frac{1 + e^{\pi s}}{1+ s^2} \cdot \frac{1}{1 - e^{-\pi s}}<br />
provided that e^{-\pi s} < 1.

    In general for infinite geometric series S = a + ar + ar^2 + ar^3 + \cdots = \frac{a}{1-r} provided that  |r| < 1.
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  5. #5
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    Thanks a lot, I appreciate it.
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