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Math Help - Countinuity of Countinued Fraction

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    Countinuity of Countinued Fraction

    Define a function,
    f(x)=[1;x,x^2,x^3,...]
    the necessary and sufficient conditions for convergence is when,
    \sum^{\infty}_{k=0}a_k diverges, thus,
    \sum^{\infty}_{k=0}x^k this is geometric.
    Diverges when |x|\geq 1 for simplicity let 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 <br />
f_n(x)=[1;x,x^2,x^3,...,x^n]<br />
which are continuous and converge pointwise to f. Prove it converges uniformly; Then the limit function is also continuous.
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