Taylor Series: What is the value of this
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Taylor Series: What is the value of this

[From: ] [author: ] [Date: 11-05-04] [Hit: ]
.. + (-1)^(n+1) * ((x-3)^2n)/n! + .. - ((x-3)^100)/50!......
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! + ((x-3)^6)/3! + ... + (-1)^(n+1) * ((x-3)^2n)/n! + .. - ((x-3)^100)/50!. What is the value of f'30(3)?

A.) -30!/15!
B.) -1/30!
C.) 1/30!
D.) 1/15!
E.) 30!/15!

Why is the answer E and not D??

-
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!.

I hope this helps!
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