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Ah yes! It actually converges to 4.

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.

**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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