Question

Taylor series question: Why is the 2 that is in a red block there? where did it come from?

Answer 1

the cos2x is expanded to 1-4x^2/2!+ ...
multiply by e^(-x)= 1 - x +x^2/2! -x^3/3! ...
the 1 from cos 2x times e^-x gives 1 - x +x^2/2! -x^3/3!
the -4x^2/2! times e^-x gives -4x^2/2! +4x^3/2! -4x^4/2! 2! ....
you get 2 x^3 terms
-x^3/3! + 4x^3/2!
or
(4/2! - 1/3!)x^3 =
(2 - 1/3!)x^3