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Math Help - summation proving

  1. #1
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    summation proving

    how do i prove
    -\displaystyle \sum_{n=-\infty}^{-1} \alpha^{n} z^{-n} =-\displaystyle \sum_{n=0}^{\infty} (\alpha^{-1}z)^{n+1}


    ???
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  2. #2
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    Quote Originally Posted by TechnicianEngineer View Post
    how do i prove
    -\displaystyle \sum_{n=-\infty}^{-1} \alpha^{n} z^{-n} =-\displaystyle \sum_{n=0}^{\infty} (\alpha^{-1}z)^{n+1}


    ???

    -\displaystyle{\sum_{n=-\infty}^{-1} \alpha^{n} z^{-n}=-\sum_{n=-\infty}^{-1}\left(\alpha^{-n} z^{n}\right)^{-1}=-\sum_{n=-\infty}^{-1}\left( \alpha^{-1} z\right)^{-n}=-\sum_{n=1}^{\infty}\left(\alpha^{-1} z\right)^{n}}

    Tonio
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  3. #3
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    Slight variation: Let j= -n. Then \alpha^n= \alpha^{-j} and z^{-n}= z^j. When n= -1, j= 1, as n goes to -\infty, j goes to \infty.

    -\sum_{n= -\infty}^{-1}\alpha^nz^{-n}= -\sum_{j= 1}^\infty\alpha^{-j}z^j= -\sum_{j= 1}^\infty \left(\alpha^{-1}z)^j and now just change the index from "j" to "n".
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