Question

The taylor series about 0 for the function f(x)=cos (x) ?

Answer 1

The coefficient is `1/2`

Explanation:

We seek the coefficient of `x^3` for the Taylor series

`f(x)=cos(x)*1/(1-x)`

Using

• `cos(x)=(1-x^2/2+x^4/24+...)`
• `1/(1-x)=(1+x+x^2+x^3+...)`

We can write this as

`f(x)=(1-x^2/2+x^4/24+...)(1+x+x^2+x^3+...)`

Notice the only terms involving `x^3`,
is obtained by `1*x^3` and `(-x^2/2)*x`

`x^3+(-x^2/2)*x=x^3-x^3/2=color(blue)(1/2)x^3`