# Math Help - summation proving

1. ## summation proving

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

???

2. Originally Posted by TechnicianEngineer
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

3. 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".