$\displaystyle \sum_{n=-\infty}^{\infty}r^{|n|}e^{inx}=$$\displaystyle \sum_{n=1}^{\infty}e^{inx}$+$\displaystyle r^{0}e^{0}$+$\displaystyle \sum_{n=-\infty}^{\infty}r^{n}e^{-inx}$

if we look at the left side

i have been told that it was broken by intervals

the left most is for n=1.. infinity the central is for n=0

the right most for n=-1 ..-infinity

i know the definition of the |x| function

for some values its x for other its -x

but here the power of e changes too

why?