Answer
$\dfrac{2}{2-x}$
Work Step by Step
The power series of $(1-x)^{-1}$ can be written as:
$1+x+x^2+x^3+......=\Sigma_{k=0}^{\infty} x^k$
Replace $x$ by $\dfrac{x}{2}$ in the above series to obtain:
$\Sigma_{k=0}^{\infty} \dfrac{x^k}{2^k}=(1-\dfrac{x}{2})^{-1}$
Therefore, the function represented by the power series is:
$(1-\dfrac{x}{2})^{-1}=\dfrac{2}{2-x}$
