1. ## maclaurin error question

Use the sum of the geometric series 1-u + u2-........ (-1)n-1un-1 to find an expression for the error when using polynomial degree n as an approximation to ln(1+ x).

well i know the gp sums to 1/(1+x)
and sum of first n terms is (1-(-u)^n))/(1+u)
when i subtract and sum -u^(n-1)+u^n+.....i get u^n/(1+u)

so integrating gives

ln(1+x)-deg(n) poly= integral {u^n/(1+u)}

but this integral is not easily done and this is high school question. so im thinking im not doing this correctly. any suggestions?

2. ## Re: maclaurin error question

Originally Posted by jiboom
Use the sum of the geometric series 1-u + u2-........ (-1)n-1un-1 to find an expression for the error when using polynomial degree n as an approximation to ln(1+ x).

well i know the gp sums to 1/(1+x)
and sum of first n terms is (1-(-u)^n))/(1+u)
when i subtract and sum -u^(n-1)+u^n+.....i get u^n/(1+u)

so integrating gives

ln(1+x)-deg(n) poly= integral {u^n/(1+u)}

but this integral is not easily done and this is high school question. so im thinking im not doing this correctly. any suggestions?

$\frac{1}{1 - (-x)} = 1 - x + x^2 - x^3 + ...$

within the interval of convergence,

$\ln(1+x) = C + x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + ...$

since $\ln(1) = 0$ , $C = 0$

$\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + ...$

the series representation is an alternating series with terms decreasing to 0.

therefore, an nth degree polynomial approximation of $\ln(x+1)$ , the magnitude of error, $E < \left| \frac{x^{n+1}}{n+1} \right|$

3. ## Re: maclaurin error question

that gives me a series for ln(1+x) and a bound for the error,but im asked to find the error when approximating ln(1+x) by using up to x^n terms in the maclaurin.

(i think this is what im being asked)

so if f(x)=x-x^2/2+....+x^n/2
i need to find the error in using f(x) to approximate ln(1+x)

so i look at ln(1+x)-f(x)

but this is the integral of {u^n/(1+u)} from my first post using
1/(1+x)=1-u+u^2+... and subtracting off 1-u+....U^(n-1)

maybe im just misunderstanding the question,but this integral can not be done by high school students.

4. ## Re: maclaurin error question

Originally Posted by jiboom
that gives me a series for ln(1+x) and a bound for the error,but im asked to find the error when approximating ln(1+x) by using up to x^n terms in the maclaurin.

(i think this is what im being asked)

so if f(x)=x-x^2/2+....+x^n/2
i need to find the error in using f(x) to approximate ln(1+x)

so i look at ln(1+x)-f(x)

but this is the integral of {u^n/(1+u)} from my first post using
1/(1+x)=1-u+u^2+... and subtracting off 1-u+....U^(n-1)

maybe im just misunderstanding the question, but this integral can not be done by high school students.
yes, you are misunderstanding the question.

you integrate $\frac{1}{1+x}$ to get $\ln(1+x)$

at the same time, you integrate the geometric series representation for $\frac{1}{1+x}$ term for term, arriving at an alternating series representation for $\ln(1+x)$.

I recommend you research the error bound for alternating series.

FYI, I teach high school, and my BC calculus students learn error bound for alternating series in addition to the Lagrange error bound.

These type problems are tested on the AP exam.

5. ## Re: maclaurin error question

thanks for your help. i think i will get a scan of the question as it has been emailed to me so some parts maybe missing.

it is valid to have the remainder term as an integral,is my integral correct ?

i agree with your post that an error bound is requried but the question as for an expression for the error.