How do I figure if a term coverges or diverges?

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?

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

Quote: ThatDonGuy

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
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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.