I am going through the paper "Moments of $P$ functions and nonclassical depths of quantum states", which contains the following passage:
A. Thermal state
The density matrix of the thermal state can be written as $$\hat{\rho}_{\text{th}} = \sum_{n=0}^{\infty}\frac{\langle n\rangle^n}{\bigl(\langle n\rangle + 1\bigr)^{n + 1}}\lvert n\rangle\langle n\rvert.\tag{3.1}$$ So, we can obtain the moments as follows: $$\begin{align}\mu_{k,l} &= \operatorname{Tr}\bigl[(\hat{a}^\dagger)^l (\hat{a})^k \hat{\rho}_{\text{th}}\bigr] \\ &= \sum_{n=0}^{\infty}\frac{\langle n\rangle^n}{\bigl(\langle n\rangle + 1\bigr)^{n + 1}}\langle n\rvert(\hat{a}^\dagger)^l (\hat{a})^k\lvert n\rangle \\ &= \sum_{n=0}^{\infty}\frac{\langle n\rangle^n}{\bigl(\langle n\rangle + 1\bigr)^{n + 1}} \frac{n!}{(n - k)!} \delta_{k,l} \\ &= k!\langle n\rangle^k \delta_{k,l}.\tag{3.2} \end{align}$$
I want to understand the last step of Eq. 3.2. How is this summation carried out to reach the final answer?