The $n$th derivative of a function $f$ at $x=0$ is given by $f^{(n)}(0)= (-1)^n \dfrac{n+1}{(n+2)2^n}$ for all $n \geq 0$. Which of the following is the Maclaurin series for $f$ ?

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The terms of the Maclaurin polynomial is given by:

$$f^{(n)}(0) \cdot \frac{x^n}{n!}$$ $$(-1)^n \dfrac{n+1}{(n+2)2^n} \cdot \frac{x^n}{n!}$$

The first three terms:

$$(-1)^0 \dfrac{0+1}{(0+2)2^0} \frac{x^0}{0!} + (-1)^1 \dfrac{1+1}{(1+2)2^1} \frac{x^1}{1!} + (-1)^2 \dfrac{2+1}{(2+2)2^2} \frac{x^2}{2!}$$ $$= \frac{1}{2}-\frac{1}{3}x + \frac{3}{32}x^2$$