$1\mathrm{.}54.$ The relations considered in this problem are used on many occasions throughout the book.

$($a$)$ Prove the validity of the following expression:

$($d$)$ Evaluate

assuming that IaI$<1.$

N$-1\{$ N a$=1$

Lan $=$ l~aN$,$ n$=$O $1-$a

for any complex number a $=$I$=1$

This is often referred to as the finite sum formula.

$($b$)$ Show that if IaI$<1,$ then

$001$ Lan$=1-$a$$

n$=$O

This is often referred to as the infinite sum formula.

$($c$)$ Show also if lal $<1,$ then

$00$

$2.$:nan $=\_\_$a$\_-=-$ n$=$O $(1-$ a$)2.$