# Continued fraction involving Fibonacci sequence

What is the limit of the continued fraction:

$$\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{5+\cfrac{1}{8+\cdots}}}}}}\$$

that involves the Fibonacci sequence terms as denominators? I've been looking for this specific continued fraction on the Internet, but I haven't been able to find anything about it. Does it converge? Is it algebraic or transcendental? Is it somehow a relevant constant? Can it be expressed in terms of $\Phi$?

• All 'regular' continued fractions are convergent and if they are also infinite, then the limit is irrational. This can be found in any (good) book on elementary number theory. Sep 3, 2018 at 19:20
• Since it's an infinite regular continued fraction it converges to an irrational number. Sep 3, 2018 at 19:20
• If you have all $1$'s as the addends, then the fraction becomes one divided by the golden ratio and its approximants are ratios of each Fibonacci number to the next one. Sep 3, 2018 at 19:21
• I tried with the Inverse Symbolic Calculator, who said "Wow, really found nothing." So it probably can't be easily expressed in terms of $\Phi$. Sep 3, 2018 at 19:26
• It is the sequence A135829(n)/A026822(n), which at least allow you to express it as a recurrence, but nothing nice seems to come from it.
– Sil
Sep 4, 2018 at 21:15

I can answer the converge aspect: this fraction is tricky - we have to be careful: $$1=\cfrac{1}{1}>\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{5+\cfrac{1}{8+\cdots}}}}}}>\frac{1}{1+something_{Big}}\approx0$$ $$\Phi>\frac35=\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{2}}}>\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{5+\cfrac{1}{8+\cdots}}}}}}>\cfrac{1}{1+\cfrac{1}{1}}=\frac12$$ $$\frac{53}{90}=\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{5}}}}}>\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{5+\cfrac{1}{8+\cdots}}}}}}>\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{3}}}}=\frac{10}{17}$$
I think we can stop here: the result $$\in(\frac{10}{17};\frac{53}{90})<\Phi$$ . Is same to say $$\in\bbox[5px,border:2px solid blue]{(\frac{900}{1530};\frac{901}{1530})}$$