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Math Help - Proving a taylor series converges.

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    Proving a taylor series converges.

    Prove the Taylor series centered around 0 converges to the indicated function.

    \ln(1 + x), x \in [0,1].
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by cgiulz View Post
    Prove the Taylor series centered around 0 converges to the indicated function.

    \ln(1 + x), x \in [0,1].
    Define "prove". Depending on the class setting your in this could be as simple as considering \int\frac{dx}{1+x} or as difficult as discussing uniform convergence.
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    I just have to show the remainder term goes to zero, i.e.,

    f(x) = T_{n}(x) + R_{n}(x).

    So either show \displaystyle\lim_{n\to\infty}T_{n}(x) = f(x),

    Or, \displaystyle\lim_{n\to\infty}\frac{f^{(n+1)}(c)}{  (n+1)!}(x - a)^{(n+1)} = 0.
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  4. #4
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    Here is what I've got so far:

    \ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - ... + (-1)^{n-1}\frac{x^n}{n} + \frac{f^{(n+1)}(c)x^{(n+1)}}{(n+1)!}.

    Now.. can I use L'Hospitals?
    Last edited by cgiulz; November 6th 2009 at 04:23 PM.
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