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Error bounds on series approximation

  1. Mar 20, 2012 #1
    1. The problem statement, all variables and given/known data

    I want to estimate [itex]f(x)=\ln (\frac{1}{1+x}) [/itex] on the interval [itex](1/10,1)[/itex] with the error on the approximation being no more than [itex]0.1[/itex].

    2. Relevant equations

    http://en.wikipedia.org/wiki/Taylor's_theorem#Example

    3. The attempt at a solution

    Following the example from Wikipedia, I first found that the [itex]n^{\text{th}}[/itex] derivative for [itex]n \geq 1[/itex] of [itex]f(x)[/itex] is:

    [tex]f^{(n)}(x)=(-1)^n \frac{(n-1)!}{(1+x)^n}[/tex]

    Then the remainder term is:

    [tex]R_{k}(x)=\frac{f^{(k+1)}(\xi_{L})}{(k+1)!}(x-a)^{k+1}=(-1)^{k+1}\frac{x^{k+1}}{(1+x)^{k+1} (k+1)}[/tex]

    Here's the sketchy part, I think I'm supposed to find an upper bound for [itex]f(x)[/itex] on the interval I want, but since [itex]f(x)[/itex] is a decreasing function, it's maximum value is going to be when [itex]x=1/10[/itex], in which case we have [itex]f(1/10) \approx -0.0953[/itex] so I guess that means I can say [itex]f(x) \leq -0.09[/itex]?

    Then:

    [tex]|R_{k}(x)| \leq \Big|(-1)^{k+1}\frac{-0.09 x^{k+1}}{(1+x)^{k+1} (k+1)}\Big| = \Big|(-1)^{k+1}\Big| \frac{|-0.09| |x|^{k+1}}{|1+x|^{k+1} |k+1|} = 0.09 \frac{x^{k+1}}{(1+x)^{k+1} (k+1)} \leq \frac{0.09}{(1+x)^{k+1} (k+1)} \leq \frac{0.09}{(k+1)} [/tex]

    Finally, impose the condition on the maximum error:

    [tex]\frac{0.09}{(k+1)} < 0.1 \implies k>-1/10 \implies k=0[/tex]

    That can't be right though. I think it might have something to do with that [itex]\xi_{L}[/itex] I disregarded.
     
    Last edited: Mar 20, 2012
  2. jcsd
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