Chapter 7.7, Problem 9E

Show that if $L\left\{g\right\}\left(s\right)={\left[\left(s+\alpha \right)\left(1-e^{-Ts}\right)\right]}^{-1}$, where $T>0$ is fixed, then

$\begin{array}{l}g\left(t\right)=e^{-\alpha t}+e^{-\alpha \left(t-T\right)}u\left(t-T\right)\\ +e^{-\alpha \left(t-2T\right)}u\left(t-2T\right)\\ +e^{-\alpha \left(t-3T\right)}u\left(t-3T\right)+\mathrm{...}\left(9\right)\end{array}$

[Hint: Use the fact that $1+x+x^2+g=1/\left(1-x\right)$.]