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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)?

Verified Answer :

syed514

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!. 

The value of  \(f^{30} (3)\) is 15!

The definition of the Taylor Series,

The coefficient of \((x - 3)^{30}\) is \(f^(30)(3) / 30!.\)

What is the formula of the Taylor series?

\(\sum \limits_{n=0}^\infty \frac {{f^{(n)}} (a)}{n!} (x-a)^n\)

n! = factorial of n

a = real or complex number

\({{f^{(n)}} (a)}\)= nth derivative of f evaluated at the point a

From the given series, it is \((-1)^(15+1) / 15!\)(letting n = 15).

Therefore, \(f^(30)(3) / 30! = (-1)^(15+1) / 15!\)

So we get,

\(f^{30}(3) = 30!/15!.\)

Therefore the value of  \(f^{30} (3)\) is 15!

To learn more about the Taylor polynomial visit:

https://brainly.com/question/23842376