Answer
$\frac{5}{2}+\frac{5x}{4}+\frac{5x^2}{8}+\frac{5x^3}{16}+...+\frac{5x^n}{2^{n+1}}+...$
Work Step by Step
We are given $f(x)=\frac{5}{2-x}=\frac{\frac{5}{2}}{1-\frac{x}{2}}=\frac{5}{2}\frac{1}{1-\frac{x}{2}}$
with $c=\frac{5}{2}$ for $r$ in $(-2,2)$
The Taylor series for $f(x)=\frac{5}{2-x}$ is
$f(x)=\frac{5}{2}.1+\frac{5}{2}(\frac{x}{2})+\frac{5}{2}(\frac{x}{2})^{2}+\frac{5}{2}(\frac{x}{2})^{3}...+\frac{5}{2}(\frac{x}{2})^n+...$
$=\frac{5}{2}+\frac{5x}{4}+\frac{5x^2}{8}+\frac{5x^3}{16}+...+\frac{5x^n}{2^{n+1}}+...$
