I have a geometric series like this :
$$\sum_{n=-\infty}^{-1}a^{-n}e^{-jwn}$$
When I make $m = -n$ substituion, it becomes this : $$\sum_{m=1}^{\infty}(ae^{jw})^m$$
And when I calculate summation, the result becomes : $ae^{jw}/(1-ae{^jw})$
I can't understand why we end up with this result. I know that geometric series solution is this :
But when I apply this formula, I can't find this result. How do we calculate the nominator of the result as ae^jw ? a and w are constants and j is the imaginary number, namely square root of -1 Thanks.
Since $\lim\limits_{b\to+\infty}r^{b+1}=0$ if $\lvert r\rvert<1$, we have: $$\lim_{b\to\infty}\sum_{k=1}^br^k=\lim_{b\to\infty}\frac{r-r^{b+1}}{1-r}=\frac r{1-r}. $$