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Math Help - maclaurin error question

  1. #1
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    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?


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  2. #2
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    Re: maclaurin error question

    Quote Originally Posted by jiboom View Post
    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|
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  3. #3
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    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.








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  4. #4
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    Re: maclaurin error question

    Quote Originally Posted by jiboom View Post
    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.
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  5. #5
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    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.
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