November 17th, 2021 at 7:59:28 AM
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I知 interested in the answer, but I知 more interested in the method for how to approach this. In the middle of a dream last night I thought of this number:

And in the dream I was trying to figure out what number it was. I knew (in the dream) that it approached a limit, but since I was dreaming it just turned into one of those circular dreams where I kept getting dumped to the beginning.

It looks like it痴 just a tick over 1/2, but how do I think about it to find out what it really is?

And in the dream I was trying to figure out what number it was. I knew (in the dream) that it approached a limit, but since I was dreaming it just turned into one of those circular dreams where I kept getting dumped to the beginning.

It looks like it痴 just a tick over 1/2, but how do I think about it to find out what it really is?

A falling knife has no handle.

November 17th, 2021 at 8:02:54 AM
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An easy one...

Let X = 1 / (1 + 1 / (1 + 1 / (1 + 1 / (....)

1 / X = 1 + 1 / (1 + 1 / (1 + 1 / (....)

1 / X - 1 = 1 / (1 + 1 / (1 + 1 / (....) = X

X - 1 / X + 1 = 0

X^2 + X - 1 = 0

X = -1/2 +/- sqrt(5)/2

but in this case, X > 0, so X = (sqrt(5) - 1) / 2

Let X = 1 / (1 + 1 / (1 + 1 / (1 + 1 / (....)

1 / X = 1 + 1 / (1 + 1 / (1 + 1 / (....)

1 / X - 1 = 1 / (1 + 1 / (1 + 1 / (....) = X

X - 1 / X + 1 = 0

X^2 + X - 1 = 0

X = -1/2 +/- sqrt(5)/2

but in this case, X > 0, so X = (sqrt(5) - 1) / 2

November 17th, 2021 at 8:07:55 AM
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Quote:ThatDonGuyAn easy one...

Let X = 1 / (1 + 1 / (1 + 1 / (1 + 1 / (....)

1 / X = 1 + 1 / (1 + 1 / (1 + 1 / (....)

1 / X - 1 = 1 / (1 + 1 / (1 + 1 / (....) = X

X - 1 / X + 1 = 0

X^2 + X - 1 = 0

X = -1/2 +/- sqrt(5)/2

but in this case, X > 0, so X = (sqrt(5) - 1) / 2

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Thanks, ThatDonGuy! I don稚 algebra much, so I had to write that down and look at it as I did it. But I got it. The method is what I was looking for. Dreams are strange.

A falling knife has no handle.