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

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.

• Are the terms in $\ldots$ repeating like $4 + 1/(\ldots)$? – Tunococ Aug 3 '13 at 2:55
• Can you at least do a search here on MSE first before ... copy and pasting your question? I'm sure (similar) questions has been asked numerous times already. – user67258 Aug 3 '13 at 2:57
• @user88786: Did you create five different accounts to ask five questions in a row? One account would work. – Amzoti Aug 3 '13 at 2:57
• It is usually a good idea to give some idea of what the $\dots$ represent. I can't tell. Is it fours all the way down? – Thomas Andrews Aug 3 '13 at 3:10

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?

• @MJD Wow thanks for that. I didn't know that command existed. – Cameron Williams Aug 3 '13 at 3:01
• Also, $y=2+\frac1{2+y}$ work as well. – user67258 Aug 3 '13 at 3:03