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

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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#