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Math Help - Series of Functions

  1. #1
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    Series of Functions

    I'll appreciate some help with this one:
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  2. #2
    Behold, the power of SARDINES!
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    Quote Originally Posted by SpinRat View Post
    I'll appreciate some help with this one:
    Hint: let

    \displaystyle g'(x)=\frac{1}{1-x^4}=\sum_{n=0}^{\infty}x^{4n}

    Now integrate term by term to get

    \displaystyle g(x)=\sum_{n=0}^{\infty}\frac{x^{4n+1}}{4n+1}=\int \frac{1}{1-x^4}dx

    Use partial fractions and integrate and let x=\frac{1}{2}
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  3. #3
    MHF Contributor chisigma's Avatar
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    Is...

    \displaystyle f^{'} (x) = 1+x^{4}+x^{8} + ... + x^{4 n} + ... = \frac{1}{1-x^{4}} = \frac{\frac{1}{2}}{1+x^{2}} + \frac{\frac{1}{2}}{1-x^{2}} (1)

    ... so that is...

    \displaystyle f(x) = \int_{0}^{x} \frac{dt}{1-t^{4}} = \frac{1}{2}\ \tan^{-1} x + \frac{1}{4}\ \ln \frac{1+x}{1-x} (2)

    ... and from (2)...

    \displaystyle \sum_{n=0}^{\infty} \frac{1}{2^{4 n}\ (4n+1)} = 2\ f(\frac{1}{2}) = \tan^{-1} \frac{1}{2} + \ln \sqrt{3} (3)

    Kind regards

    \chi \sigma
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