Question$-4$:

Prove the following expression $(10$ points$)$

${\displaystyle \underset{n=0}{\overset{N-1}{}}}{}^n=\{\begin{array}{cc}\frac{1-{}^N}{1-},& 1\\ N,& =1\end{array}$

And find the result of following expression for $|0|<1$

$(5$ points$)$

${\displaystyle \underset{n=0}{\overset{}{}}}{}^n=$