$(25$ points$)$ Find the Radius and interval of convergence of the power series: ${\displaystyle \underset{n=0}{\overset{}{}}}(\frac{x^n}{2^n}),$ where $x$ is a real number.

After your complete work and finding the answer, choose the right answer from the following possible answers:

a$.$ Radius $of$ convergence :$R=1$

Interval $of$ convergence: $(-1,1)$

b$.$ Radius $of$ convergence: $R=2$

Interval $of$ convergence: $(-2$;$2)$

c Radius $of$ convergence : $R=1$

Interval $of$ convergence: $(-2\mathrm{.}2)$

d$.$ None of the above