sequences and series - Sum of $sum_{k=0}^{infty} k rho^{k-1}$

I encounter the infinite series $\sum_{k=0}^{\infty} k \rho^{k-1}$ in a math textbook where the answer is directly given to be $\dfrac{1}{(1-\rho)^2}$ when $|\rho| < 1$. However, I don't understand how to obtain this result. Could someone give me a hint of how to approach this problem?

Answer

Hint: Think of the convergent series $$\sum_{k=0}^\infty \rho^k=\frac{1}{1-\rho}, \qquad |\rho|<1,$$ and differentiate both sides with respect to $\rho$ (why is interchanging differentiation and summation possible?).