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Math Help - Power series expansion

  1. #1
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    Power series expansion

    Expand power series of 1/(cosx) analytic at x=0. Find the first 4 terms of the series.
    I know the taylor series for cosx, but isn't this just the same thing but with all the fractions flipped?
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  2. #2
    MHF Contributor chisigma's Avatar
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    Lets suppose that the McLaurin expansion of \frac{1}{\cos x} exists and write...

    \frac{1}{\cos x} = \sum_{n=0}^{\infty} a_{n}\cdot x^{n} (1)

    In this case we can derive the a_{n} from the McLaurin expansion of \cos x...

    \cos x = \sum_{k=0}^{\infty} b_{k}\cdot x^{k} (2)

    ... where b_{k} = \frac{(-1)^{\frac{k}{2}}}{k!} for k even and b_{k}=0 for k odd. If we write the product of the series (1) and (2) we obtain...

    \sum_{n=0}^{\infty} a_{n}\cdot x^{n} \cdot \sum_{k=0}^{\infty} b_{k}\cdot x^{k} = \sum_{n=0}^{\infty} x^{n}\cdot \sum_{k=0}^{n} a_{k}\cdot b_{n-k} = 1 (3)

    ... and from (3) we obtain...

    a_{0}\cdot b_{0}= 1 \rightarrow a_{0} = 1

    a_{0}\cdot b_{1} + a_{1}\cdot b_{0} = 0 \rightarrow a_{1} = 0

    a_{0}\cdot b_{2} + a_{1}\cdot b_{1} + a_{2}\cdot b_{0} = 0 \rightarrow a_{2}= \frac{1}{2}

    a_{0}\cdot b_{3} + a_{1}\cdot b_{2} + a_{2}\cdot b_{1} + a_{3}\cdot b_{0} = 0 \rightarrow a_{3} = 0

    a_{0}\cdot b_{4} + a_{1}\cdot b_{3} + a_{2}\cdot b_{2} + a_{3}\cdot b_{1} + a_{4}\cdot b_{0} = 0 \rightarrow a_{4} = - \frac{5}{24}

    The McLaurin expansion of \frac{1}{\cos x} with maximum exponent 4 is then...

    \frac{1}{\cos x} = 1 + \frac{1}{2}\cdot x^{2} - \frac{5}{24}\cdot x^{4} + \dots (4)

    Kind regards

    \chi \sigma
    Last edited by chisigma; March 4th 2010 at 01:31 PM.
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