pull down to refresh

[Daily puzzle] Exact value of continued fraction
634 sats \ 7 comments \ @south_korea_ln 6 Oct science
After a brief hiatus, let's pick up with our little daily puzzles.
Showcasing the power of the recent addition of MathJax support on SN (thank you for that), let's start with a little continued fraction puzzle.
Can you find the exact value of x in this continued fraction?
FYI, here the corresponding syntax:
$$
x = 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \cdots}}}}
$$
Previous iteration: #706302
write
preview
related posts
200 sats \ 5 replies \ @SimpleStacker 6 Oct
x is the solutions to the quadratic equation above.
reply
155 sats \ 4 replies \ @south_korea_ln OP 7 Oct
That is correct... with a small caveat on the fact 2 solutions for the quadratic equation but only one of them is valid for the continued fraction.
Bonus question, what is special about the solution to this continued fraction?
reply
0 sats \ 3 replies \ @SimpleStacker 7 Oct
Oh, interesting. Why is one of the quadratic solutions not a solution? I assume the negative one is the one that’s not a solution, maybe because the fraction has to stay positive?
I didn’t realize it at first, but now that you mention it’s special, I see that it’s the golden ratio!
reply
0 sats \ 2 replies \ @south_korea_ln OP 7 Oct
the fraction has to stay positive?
Yes, it's hard for this infinite continued fraction to ever turn positive ;)
it’s the golden ratio!
Yes indeed :)
reply
0 sats \ 1 reply \ @SimpleStacker 7 Oct
By the way, I was wondering about this proof method. Would the following proof work?
To shed some light, I'm an economist, not a mathematician. But I have used this method to prove the formula for a discounted stream of infinite cash flows:
I guess it has to do with whether a series converges or not. If it doesn't you probably can't assume that , or something like that.
reply
0 sats \ 0 replies \ @south_korea_ln OP 7 Oct
Yes, this seems like a similar approach to prove your expression, replacing part of the infinite expression with itself. Not familiar with the economics part of it though ;)
reply
0 sats \ 0 replies \ @south_korea_ln OP 7 Oct
@SimpleStacker gave the right answer.
If you don't see how they got the quadratic equation, notice that this is a infinite continued fraction. So, everything below the first numerator (i.e. the infinitely long denominator) is equal to $x$ itself. Hence, $x = 1+1/x$.
To solve the quadratic equation, use the usual $\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ formula, and discard the negative solution as explained in another comment. We then get $\frac{1+\sqrt{5}}{2}\simeq 1.618$ which is the golden ratio.
reply