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#1 Re: This is Cool » Another phi curiosity » 2011-04-24 03:04:08

Ah yes! It actually converges to 4.

#2 Re: This is Cool » Another phi curiosity » 2011-04-23 18:29:35

Hi bobbym,

Thanks for the reply. The surprise for me is all those odd powers of an irrational conspiring to yield an integer! Putting the next phi-like term in the series, that is (3 + √5)/2, for n leaves the convergence unchanged at 2 x phi but now yields 3 when truncated to only 2 terms, leaving a somewhat more trivial algebra of (3 + √5)/2 + 2/(3 + √5) = 3.

#3 This is Cool » Another phi curiosity » 2011-04-23 01:28:09

UltraGnosis
Replies: 5

I was playing around with the series n + 1/n + 1/n² + 1/n³... in a spreadsheet and noted that it converges to n + 1/(n-1) for n > 0. Putting n = phi obviously gives the convergence of 2 x phi. However, stopping the sum at the 5th term, I was surprised to see that phi + 1/phi + 1/phi^2 + 1/phi^3 + 1/phi^4 = 3 exactly (well, to 14 places at least). I'm wondering whether:

a) Anyone's ever noted this before
b) There's anything deeper here, such as a geometric explanation

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