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Nothing's perfect. (Posted on 2016-12-24) Difficulty: 3 of 5

The bisection of the Fibonacci series, Sloane A001906 {1, 3, 8, 21, 55, 144,...}, naturally produces approximations to phi^2, by the division of the nth term by its predecessor: a(n)/a(n-1). ; e.g 55/21, 144/55, etc.

WolframAlpha also lists these fractions as convergents to 5pi/6.

In fact, there will always be a small shortfall between the two: (phi)^2 - 5pi/6 is not zero.

For sufficiently large n, how is the shortfall best approximated, in terms of a rational fraction, say 1/x?

See The Solution Submitted by broll    
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  Subject Author Date
Some Thoughtscomputer explorationCharlie2016-12-26 08:56:34
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