APFIVE Coefficient of a Composite Maclaurin Series
A function $$f$$ has a Maclaurin series given by $$1 + 2x + \displaystyle\frac{1}{2}x^2 + \displaystyle\frac{1}{6}x^3 + \cdots$$, and the Maclaurin series converges to $$f(x)$$ for all real numbers $$x$$. If $$g$$ is the function defined by $$g(x) = e^{f(x)}$$, what is the coefficient of $$x^3$$ in the Maclaurin series for $$g$$?
A
$$\displaystyle\frac{5}{2} e$$
B
$$\displaystyle\frac{3}{2} e$$
C
$$\displaystyle\frac{7}{3} e$$
D
$$\displaystyle\frac{8}{3} e$$
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