Example 4.24 Find the exponential generating function of the sequence $\left\{r^{n}\right\}_{n \geqslant 0}$, where $r$ is a positive real number.
Solution: Since $\sum_{n=0}^{\infty} r^{n} t^{n} / n!=\sum_{n=0}^{\infty}(r t)^{n} / n!=\mathrm{e}^{r t}$, the exponential generating function of the sequence $\left\{r^{n}\right\}_{n \geqslant 0}$ is $\mathrm{e}^{r t}$.