Mathematics

This is a power series problem where you need to recognize that the sum of a geometric series and its derivative are closely related. Start by recalling that $\sum_{k=0}^{\infty} x^k = \frac{1}{1-x}$ for $|x| < 1$, then differentiate both sides with respect to $x$ to get a formula for $\sum_{k=1}^{\infty} k x^{k-1}$. From there, a small algebraic adjustment will give you exactly the form you need, with $x = \frac{1}{4}$.

This is a power series problem where you need to recognize that the sum of a geometric series and its derivative are closely related. Start by recalling that $\sum_{k=0}^{\infty} x^k = \frac{1}{1-x}$ for $|x| < 1$, then differentiate both sides with respect to $x$ to get a formula for $\sum_{k=1}^{\infty} k x^{k-1}$. From there, a small algebraic adjustment will give you exactly the form you need, with $x = \frac{1}{4}$.