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Math Help - Express as power series

  1. #1
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    Express as power series

    Is it possible to express \frac{1}{e^t+e^{-t}} as a power series in e^{-t}?

    So can we find a_n with \frac{1}{e^t+e^{-t}} = \sum_{n=0}^\infty a_n (e^{-t})^n ?
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  2. #2
    MHF Contributor chisigma's Avatar
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    Setting e^{-t} = x we obtain...

    \frac{1}{e^{t} + e^{-t}} = \frac{1}{x+\frac{1}{x}} = \frac{x}{1+x^{2}} =

    = x\cdot (1-x^{2} + x^{4} - \dots) = x - x^{3} + x^{5} - \dots (1)

    ... so that is...

    \frac{1}{e^{t} + e^{-t}} = e^{-t} - e^{-3t} + e^{-5t} - \dots (2)

    Because the series (1) converges for |x|<1, the series (2) converges for t>0 ...

    Kind regards

    \chi \sigma
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  3. #3
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    Didn't we just have this exact problem recently?
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