Using the fact that ((1 / n !)) is the coefficient of (x^{n}) in the power expansion

Question:

Using the fact that $(1 / n !)$ is the coefficient of $x^{n}$ in the power expansion of the function $\exp (x)$, derive an asymptotic formula for this coefficient by the method of saddle-point integration. Compare your result with the Stirling formula for $n$ !.

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