2016 AMC 10B Problem 16

Let the first value of the series be $a,$ and let the ratio be $r.$ Thus, $$\begin{aligned}S&=\dfrac{a}{1-r}\\ &= \dfrac{ar}{r(1-r)} \\&= \dfrac{1}{r(1-r)}.\end{aligned}$$ This means we have to find $r$ that maximizes $$r(1-r)= 0.25-(r-0.5)^2.$$ This maximization will happen with $r=0.5.$ Therefore, $S = \dfrac{1}{0.5(0.5)}= 4.$ Thus, the correct answer is E .