Infinitely Nested Radical with Fibonacci Coefficients

Asked Mar 05 '19 at 16:02

Active Dec 05 '23 at 10:29

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I was wondering if the following infinitely nested radical can be evaluated.

$$x= \sqrt{1+ \color{red}{1}\sqrt{1+ \color{red}{1}\sqrt{1+ \color{red}{2}\sqrt{1+ \color{red}{3}\sqrt{1+ \color{red}{5}\sqrt{1+ \dots }}}}}} \,= \,??$$

with coefficients being the Fibonacci numbers. Generally, is there a way to evaluate all infinitely nested radicals with coefficients being the terms of a fibonacci-like sequence?

Thank you!

edited Dec 05 '23 at 10:29

Tito Piezas III

54,565

asked Mar 05 '19 at 16:02

Altun Hasanli

1

I have seen a solution on this site. Start here, or here, or here from Ramanujan's note book... – Dietrich Burde Mar 05 '19 at 16:11
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Does this answer your question? Evaluating the nested radical $ \sqrt{1 + 2 \sqrt{1 + 3 \sqrt{1 + \cdots}}} $. – Lee David Chung Lin Feb 11 '20 at 03:31
@DietrichBurde I don't think Ramanujan considered this case. Ramanujan's "coefficients" for his famous nested radical were $2,3,4,\dots$, while Altun used the Fibonacci numbers. – Tito Piezas III Dec 05 '23 at 10:34
Can anyone find the value of $x$ to 10 or more digits? – Tito Piezas III Dec 05 '23 at 10:36

Infinitely Nested Radical with Fibonacci Coefficients

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