Visualisation of the reciprocal of an continued fraction?

Question

If: $$a=\cfrac{l}{m+\cfrac{n}{o+\cfrac{p}{q+\cdots}}}$$

Then could you help me visualize $1/a$? I really don't understand it. Thank you so much!

Answer 1

Your continued fraction is very generic, in fact it specifies only $c_0 = 0$ and $b_1 = 1$ (compare with Batominovski's expansion) the general form $$ x = \frac{1}{a} = c_0 + \cfrac{b_1}{c_1+\cfrac{b_2}{c_2+\cfrac{b_3}{c_3+\cdots}}} $$

How to visualize such a limit process?

For specific values of the coefficients $a_i$ and $b_i$ one might plot the sequence of the fraction values \begin{align} x_0 &= c_0 \\ x_1 &= c_0 + \cfrac{b_1}{c_1} \\ x_2 &= c_0 + \cfrac{b_1}{c_1 + \cfrac{b_2}{c_2}} \\ & \vdots \end{align} for increasing index $i$.

If the coefficients are repetitive, e.g. $b_i = b$, $c_i = c$, then we can express the values as \begin{align} x_1 &= \phi(c) = \phi^1(c) \\ x_2 &= \phi(\phi(c)) = \phi^2(c) \\ x_3 &= \phi(\phi(\phi(c))) = \phi^3(c) \\ & \vdots \end{align} with $$ \phi(x) = c + \frac{b}{x} $$ For specific values of $b$ and $c$ we could plot this as a fixed point iteration.

(Larger image version)

The above image shows the iteration for $c=2$ and $b=1$ which converges to $1+\sqrt{2}$.

Answer 2

What about $\cfrac{1}{a}=\cfrac{1}{0+\cfrac{l}{m+\cfrac{n}{o+\cfrac{p}{q+\ldots}}}}$? In fact, I don't know what you mean by "visualization."