The sum of infinite series $1 - \frac{2^2}{5} - \frac{3^2}{5^2} - \frac{4^2}{5^3} - \frac{5^2}{5^4} - \frac{6^2}{5^5} - \ldots$ is equal to:
Step-by-Step Solution
Key Concept: Differentiate the geometric series formula twice to convert a power series with coefficients into a rational function.
Given $S = \sum_{r=1}^{\infty} r^2 \left( rac{1}{5}
ight)^{r-1}$. Starting from the geometric series $\sum_{r=0}^{\infty} x^r = rac{1}{1-x}$, differentiate twice to get $\sum_{r=1}^{\infty} r^2 x^{r-1} = rac{1+x}{(1-x)^3}$. Substituting $x = rac{1}{5}$ yields $S = rac{1 + rac{1}{5}}{(1- rac{1}{5})^3} = rac{ rac{6}{5}}{( rac{4}{5})^3} = rac{25}{8}$.
Correct Answer: 4