find the value of $\frac{1}{2+\frac{1}{4+\frac{1}{4+\frac{1}{\ldots}}}}$

Question

the question is to find the value of this ugly non-stopping fraction $$\frac{1}{2+\frac{1}{4+\frac{1}{4+\frac{1}{\ldots}}}}$$.

I have totally no clue; thanks for the help! How am I suppose to solve this thing? This thing certainly looks ugly.

Thank you for all the help.

Answer 1

Let $x$ be your continued fraction so that:

$$x = \cfrac{1}{2+\cfrac{1}{4+\cfrac{1}{4+\cdots}}}.$$

Notice the repeating nature after a while in the continued fraction (this is very important). We want to make use of this. To do so let's invert both sides:

$$x^{-1} = 2 + \cfrac{1}{4+\cfrac{1}{4+\cfrac{1}{4+\cdots}}}.$$

Or written a slightly different way..

$$x^{-1} - 2 = \cfrac{1}{4+\cfrac{1}{4+\cfrac{1}{4+\cdots}}}.$$

We are now left with evaluating the continued fraction on the right. This is actually simpler than it looks because of the self-similar behavior.

Define the following:

$$y = \cfrac{1}{4+\cfrac{1}{4+\cfrac{1}{4+\cdots}}}.$$

Then:

$$y^{-1} = 4 + \cfrac{1}{4+\cfrac{1}{4+\cfrac{1}{4+\cdots}}}.$$

Or..

$$y^{-1} = 4 + y.$$

Can you solve this for $y$? Do you see how this solves the problem at hand?