Consider the series ${\displaystyle \underset{k=0}{\overset{}{}}}\frac{k^2}{5^k}.$

$($a$)$ In $1-2$ sentences, explain why this series is NOT geometric.

$($b$)$ Use one of the series tests to confirm that this series converges.

$($c$)$ Use the power series ${\displaystyle \underset{k=0}{\overset{}{}}}{x}^k=\frac{1}{1-x},|x|<mn>1</mn>$ and find a power series representation of ${\displaystyle \underset{k=0}{\overset{}{}}}{k}^2{x}^k.$

Hint: You should perform some differentiation and multiplication of monomials on the given power series.

$($d$)$ Find the interval of convergence of your power series representation in part$($b$).$

$($e$)$ Take the power series you found in part $($b$)$ and plug in some value for $x$ inside the interval of convergence to find the sum of the series given at the beginning of this problem.