A function $f$ has Maclaurin series given by $1+\frac{x^2}{2!}+\frac{x^4}{4!}+\frac{x^6}{6!}+...+ \frac{x^{2n}}{(2n)!} + ...$. Which of the following is an expression for $f(x)$ ?

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The expansion of $e^x$ is:

$$1+x+\frac{x^2}{2!}+\frac{x^3}{3!} + ...$$

If we want only the even powers, we can add the expansion of $e^{-x}$:

$$1-x+\frac{x^2}{2!}-\frac{x^3}{3!} + ...$$

Adding these two expansions result in:

$$2+\frac{2x^2}{2!}+\frac{2x^4}{4!}+...$$

Dividing this by $2$ results in the requested series.

$$\frac{e^x-e^{-x}}{2}$$