# What are the first 3 nonzero terms in the Taylor series expansion about x = 0 for the function f(x)=cos(4x)?

##### 1 Answer
Mar 17, 2015

Answer. $1 - 8 {x}^{2} + \frac{32}{3} {x}^{4}$.

Because

$\cos \left(X\right) = 1 - {X}^{2} / 2 + {X}^{4} / 24 + {o}_{X \to 0} \left({X}^{4}\right)$,

and because $4 x \to 0$ when $x \to 0$, you get

$\cos \left(4 x\right) = 1 - {\left(4 x\right)}^{2} / 2 + {\left(4 x\right)}^{4} / 24 + {o}_{x \to 0} \left({\left(4 x\right)}^{4}\right)$,

so

$\cos \left(4 x\right) = 1 - 8 {x}^{2} + \frac{32}{3} {x}^{4} + {o}_{x \to 0} \left({x}^{4}\right)$,