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

enter image source here

1 Answer
Mar 12, 2018

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