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155 sats \ 4 replies \ @south_korea_ln OP 7 Oct \ parent \ on: [Daily puzzle] Exact value of continued fraction science
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?
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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