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I read that it was proved that reciprocal Fibonacci constant $$\sum_{n} \frac{1}{F_n} = \frac{1}{1} + \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{5} + \frac{1}{8} + \frac{1}{13} + \frac{1}{21} + \cdots \approx 3.3598856662 \dots .$$ is irrational.

Can anyone show me the proof or is it too difficult for someone who knows basic number theory?

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I can't read french. –  user58512 Jan 22 '13 at 19:08

1 Answer 1

up vote 3 down vote accepted

In Matala-Aho and Prevost, "Quantitative irrationality for sums of reciprocals of Fibonacci and Lucas numbers" (http://cc.oulu.fi/~tma/TAPANI20.pdf) a generalization of this result is presented.

The material is quite technical. If you aren't familiar with Pade Approximants or Cyclotomic Polynomials it will be tough to follow.

The general idea is to replace such a sum with a a function, say $f(t)$, and to recover the sum as $f(1)$. The majority of the work is to provide the technical machinery to support inequality (4). This inequality gives an explicit bound on how "close" a rational number can come within a point of interest. With good enough bounds, this is used to show that a value is irrational.

Hope this helps.

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thanks, is this harder than the proof for just this one series? –  user58512 Jan 22 '13 at 19:28
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The generalized version uses the same methods as the proof for just your series. I would say they are of roughly equal difficulty. –  pre-kidney Jan 22 '13 at 19:42

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