I found then that if you iterate a geometric series on itself and then take the number of iterations to infinity you obtain... φ-1 
pretty neat..pic.twitter.com/dgtFoPFOBE
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This is definitely one of the coolest things I think I've ever figured out..
For some reason, i decided to take the geometric sum (S₁) of a geometric series (S)
Then i decided to take the sum of *that* geometric series (S₂)
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Then a familiar pattern started to emerge...pic.twitter.com/CxFxPWJw8O
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I think this is geometrically the fact that there is only one spiral that does like the perfect spiralling down to a point. <waves hands>
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What do you mean exactly? There’s many many spirals
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You can write the recurrence relation. At the limit the n+1 term is the same as the nth term and you have the quadratic that solves to give the golden ratio. Also rewriting Sn as a/b you get Sn+1=b/(a+b) which can describe the ratios between successive Fibonacci sequence terms
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I should say Fibonacci like sequence. The main thing is the recurrence relation. Nice observation though. :)
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Nice these continued fractions analysis.
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Please, give me a case where |r|<1=>|S|<1
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one of those results where your mouth stays open for a while
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Why the minus sign?
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