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Nils-Ake

Member

Registered: 2008-04-15

Posts: 20

Taylor polynomial

Hi. I'm doing something wrong here but I can't find out what it is.

Use Taylor's theorem to determine the degree of the Taylor polynomial for ln(1-x) required for the error in the approximation of ln(0.75) to be less than .001.

(I assume it is centered at a = 0)

Derivatives:

Taylor polynomial:

Where

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Now I evaluate the polynomial for x=0.25

At this point I plotted f^n+1 and saw that

So this should give me the answer

I get n>=3 but the book says n>=4. Where did I go wrong?

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Taylor polynomial

Hi Nils-Ake;

I do not know exactly where you went wrong but the book answer is right. Here is how I would compute the error.

a = 0
x = .25

a ≤ z ≤ x

z is picked to make the fourth derivative a maximum. Compute the fourth derivative.

Just substitute now. Use z = .25 because that makes the fourth derivative a maximum in the interval of 0 and .25.
I use the absolute value of the derivative.

Use the absolute value.

Solve for n by trying n=1, n=2, n=3, n=4 etc.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
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