A lvl H2 Maths: Maclaurin's Series

This is on Maclaurin series

Find f(x), whose expansion is given by
$f(x)=2+x+\frac{x^{3}}{3!}+\frac{2x^{4}}{4!}+\frac{x^{5}}{5!}+\frac{x^{7}}{7!}+\frac{2x^{8}}{8!}+...$

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Answer:

$f(x)=2+x+\frac{x^{3}}{3!}+\frac{2x^{4}}{4!}+\frac{x^{5}}{5!}+\frac{x^{7}}{7!}+\frac{2x^{8}}{8!}+...$

$f(x)=\left( 1+x+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+\frac{x^{4}}{4!}+\frac{x^{5}}{5!}+\frac{x^{6}}{6!}+\frac{x^{7}}{7!}+\frac{x^{8}}{8!}+...\right)$
$+\left( 1-\frac{x^{2}}{2!}+\frac{x^{4}}{4!}-\frac{x^{6}}{6!}+\frac{x^{8}}{8!}+...\right)$

$\large \mathbf{f(x)=e^{x}+\cos x}$ (done)

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A lvl H2 Maths: Maclaurin's Series