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