Chapter 5, Problem 5.19TE

If X is an exponential random variable with mean $\frac{1}{\lambda }$, show that $E\left[X^k\right]=\frac{k!}{{\lambda }^k}k=1,2,\mathrm{...}$

Hint: Make use of the gamma density function to evaluate $E\left[X^k\right]$.