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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
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
For what values of r does S_k converge? I'm guessing it's something like k/(k+1)<r<1 which is a pretty skinny interval as k increases!
it’s actually 0<r<1 ! if r>0 then the result of the next iteration will be smaller!
Something probably relevant... Phi-1=1/phi so this is the reciprocal of phi :)
φ depends of something?
This phi?
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