user12692
0
A common "trick" for obtaining a closed form of a geometric series is to define
$$ R := \sum_{k=0}^{\infty} r^k, $$
then manipulate the series as follows:
\begin{align}
R - rR
&= \sum_{k=0}^{\infty} r^{k} + \sum_{k=0}^{\infty} r^{k+1} \\
&= (1 + r + r^2 + r^3 + \dotsb) - (r + r^2 + r^3 + \dotsb) \\
