Maclaurin polynomial

zhupolongjoe · #1 $Old$ November 2nd, 2009, 07:12 PM

Consider the Maclaurin polynomial of degree N for f(x)=e^x.

i---Give an estimate for the error term when we use this to approximate
e^(-1)

ii---For what N will the Maclaurin polynomial at -1 give an approximationg for e^(-1) accurate to 6 decimal places. Compute this approximation.

Well, I know the polynomial is 1+x+x^2/2+x^3/3!+...+x^N/N!

For part i, do you just do 1/e-(1+(-1)+1/2-1/3+......+(-1)^N/N!)
Is this the estimate of the error term? I'm not sure how you do this or part 2?

zhupolongjoe · #2 $Old$ November 2nd, 2009, 08:35 PM

Okay, for ii, I found n=9 just by trial and error, but how could I have gotten this mathematically?

HallsofIvy · #3 $Old$ November 3rd, 2009, 04:48 AM

There is a formula, which I suspect you were expected to know, that says the error, in cutting off a Maclaurin series of a function at the nth power, is less than or equal to $\frac{M}{n!}|x|^n$ where "M" is an upper bound on the n+1 derivative of the function between 0 and x.

Here, since every derivative is again $e^{x}$ and that has a maximum of 1 on any interval [-x, 0], that maximum error is $\frac{1}{n!}< .000001$ . That is the same as saying that $n!> 1000000$
8!= 40320< 1000000 and 9!= 362880> 1000000 so you need 9 terms.