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$$A = a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{a_4 + \cdots}}},$$ where $a_n = f(n), f:\mathbb{Z^+} \rightarrow \mathbb{Z^+}$.

Is there an easy way of calculating the value of $A$ using algebra and/or simple calculus for a given $f(n)$? And are all such fractions convergent?

Michael Hardy
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Tahir Imanov
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  • All such regular continued fractions are convergent. I doubt there's a simple way to compute the value for an arbitrary $f$, but I'm no expert. There's a beautiful book by Khinchin on the subject. – saulspatz Oct 19 '19 at 18:07
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    The only real way of computing $A$ is available when the sequence $a_n$ is (eventually) periodic. Apart from that, very rarely does $A$ have any reasonable closed form. To showcase the complexity of what we are dealing with, for $a_n=n$ the value of $A$ requires Bessel functions to express, see here – Wojowu Oct 19 '19 at 18:07

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