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Math Help - Taylor Series

  1. #1
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    Taylor Series

    I'm suppose to use the Taylor series at x = 0 for
     <br />
\frac{1}{1-x} which, I believe, is 1 + x + x^2 + x^3 + ..., <br />

    for the following function:  \frac{x}{1 + x^3}

    I have no idea how I 'apply' the first to the second. What does that mean?

    What does  <br />
\frac{1}{1-x} = 1 + x + x^2 + x^3 + ..., <br />

    have to do with the second function and how do I 'apply' it?
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  2. #2
    MHF Contributor chisigma's Avatar
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    Why don't write in the expression \frac{1}{1-x} instead of x [for example...] -x^{3} ?...

    Kind regards

    \chi \sigma
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  3. #3
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    I don't understand, what do I replace with what, and where, and why?

    (I'm still confused).
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  4. #4
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    you know that \frac1{1-x}=\sum x^n which is the Taylor series at x=0 (well known as McLaurin series), then put x\mapsto -x^3, thus \frac1{1+x^3}=\sum (-1)^nx^{3n} so multiply both sides by x and you're done.
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  5. #5
    MHF Contributor chisigma's Avatar
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    For |x|<1 is...

    \frac{1}{1-x} = 1 + x + x^{2} + \dots (1)

    If we replace in (1) x with -x^{3} we obtain...

    \frac{1}{1+x^{3}} = 1 - x^{3} + x^{6} - \dots (2)

    ... and now we multiply both terms of (2) by x we obtain...

    \frac{x}{1+x^{3}} = x - x^{4} + x^{7} - \dots (3)

    Kind regards

    \chi \sigma
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