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Argumenting for this inequality involving the residual of a taylor series of ln

  1. Oct 7, 2012 #1

    lo2

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    1. The problem statement, all variables and given/known data

    OK I have to argument for the fact that this inequality is true, where x > 1.

    [itex] |R_n \ln{x}| \leq \frac{1}{n+1}(x-1)^{n+1}[/itex]

    And I have found out that the residual is equal to this:

    [itex]R_n \ln{x} = \frac{1}{n!} \int^x_a{f^{n+1}(t)(x-t)^{n}dt}[/itex]


    2. Relevant equations



    3. The attempt at a solution

    So since I have to just argument, not prove.

    Can I just look at the parts [itex]\frac{1}{n!}[/itex] and [itex](x-1)^{n+1}[/itex], and then say that the first one goes towards zero very fastly as n increases whereas the other one goes towards infinity quite rapidly as well. And therefore this inequality must be true.

    Also I perhaps should add that this part:

    [itex] \frac{(x-1)^{n+1}}{n+1}[/itex]

    Goes toward infinity as the exponential function out-sprints the linear function. Of course only for x > 2.
     
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