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

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

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: $(-2,2).$

d$.$ Neee of the above