# Countinuity of Countinued Fraction

• Feb 27th 2006, 04:15 PM
ThePerfectHacker
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 :(
• Aug 26th 2006, 12:44 PM
Rebesques
Take the sequence $
f_n(x)=[1;x,x^2,x^3,...,x^n]
$
which are continuous and converge pointwise to $f$. Prove it converges uniformly; Then the limit function is also continuous.