Question

The Taylor polynomial of degree 100 for the function f about x=3 is given by

\[p(x)= (x-3)^2 - ((x-3)^4)/2! +... + [(-1)^n+1] [(x-3)^n2]/n! +... - ((x-3)^100)/50!]/
What is the value of f^30 (3)?

Answer 1

By the definition of the Taylor Series, the coefficient of (x - 3)^30 is f^(30)(3) / 30!.

On the other hand, looking at the given series, it is (-1)^(15+1) / 15! (letting n = 15).

Hence, f^(30)(3) / 30! = (-1)^(15+1) / 15!
==> f^(30)(3) = 30!/15!.