Example 4.2 Let there be a formal power series $A(t)=\sum_{n=0}^{\infty} \frac{1}{n!} t^{n}$, find $A^{k}(t)$.
\[
\begin{aligned} 
A^{k}(t) & =\left(\sum_{n=0}^{\infty} \frac{1}{n!} t^{n}\right)^{k}=\left(\mathrm{e}^{t}\right)^{k}=\mathrm{e}^{k t} \\ 
& =\sum_{n=0}^{\infty}(k t)^{n} / n!=\sum_{n=0}^{\infty} \frac{k^{n}}{n!} t^{n} .
\end{aligned}
\]