Example 4.3 Find the inverse of the formal power series $A(t)=\sum_{n=0}^{\infty} \frac{1}{n!} t^{n}$.

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Solution: Since $A(t)=\sum_{n=0}^{\infty} \frac{1}{n!} t^{n}=\mathrm{e}^{t}$,
thus $A^{-1}(t)=\mathrm{e}^{-t}=\sum_{n=0}^{\infty} \frac{(-1)^{n}}{n!} t^{n}$