Let $x$ be the sum of the series $1+2r+3r^2+\dots$ to infinity, where $|r|<1$. If $x=\frac{9}{4}$, find $r$.

Solution

1. The sum of the infinite series $1+2r+3r^2+\dots$ is given by the formula $x=\frac{1}{(1-r{)}^2}$ for $|r|<1$.
2. Given $x=\frac{9}{4}$, we set up the equation $\frac{1}{(1-r{)}^2}=\frac{9}{4}$.
3. Taking the reciprocal of both sides, we get $(1-r{)}^2=\frac{4}{9}$.
4. Taking the square root of both sides, we have $1-r=\frac{2}{3}$, which simplifies to $$\begin{array}{rl}r& =1-\frac{2}{3}\\ & =\frac{1}{3}\end{array}$$.
5. Hence the answer is (A).