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0 sats \ 3 replies \ @SimpleStacker 7 Oct 2024 \ parent \ on: [Daily puzzle] Exact value of continued fraction science
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!
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 :)
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By the way, I was wondering about this proof method. Would the following proof work?
x = 1 - (1 - (1 - (1 - \dots x = 1 - xx = 0.5To 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:
NPV = 1 + \delta + \delta^2 + \ldotsNPV = 1 + \delta (1 + \delta + \ldots NPV = 1 + \delta NPVNPV = \frac{1}{1-\delta}I guess it has to do with whether a series converges or not. If it doesn't you probably can't assume that
x \in \mathbb{R}, or something like that.reply
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 ;)
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