200 sats \ 5 replies \ @SimpleStacker 6 Oct \ on: [Daily puzzle] Exact value of continued fraction science
x is the solutions to the quadratic equation above.
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?
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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!
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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?
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
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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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