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) ?

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!.