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Thread: Countinuity of Countinued Fraction

  1. #1
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    Countinuity of Countinued Fraction

    Define a function,
    $\displaystyle f(x)=[1;x,x^2,x^3,...]$
    the necessary and sufficient conditions for convergence is when,
    $\displaystyle \sum^{\infty}_{k=0}a_k$ diverges, thus,
    $\displaystyle \sum^{\infty}_{k=0}x^k$ this is geometric.
    Diverges when $\displaystyle |x|\geq 1$ for simplicity let $\displaystyle x\geq 1$. Now prove that this series is countinous.

    I am trying to express transcendental functions in terms of infinite countinous fractions, like the one above. I do not think I will get anywhere
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    Super Member Rebesques's Avatar
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    Take the sequence $\displaystyle
    f_n(x)=[1;x,x^2,x^3,...,x^n]
    $ which are continuous and converge pointwise to$\displaystyle f$. Prove it converges uniformly; Then the limit function is also continuous.
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