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Can every generalised continued fraction be written in simple continued fraction form? If not, how else does one represent a generalised continued fraction using the standard $[a_0;a_1,a_2,\ldots]$ notation when there are $b_1,b_2,\ldots$ to consider?

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  • $\begingroup$ Every real number can be written as s simple continued fraction, so yes , assuming the original generalized continued fraction converges towards a real number. There should even be rules for the conversion. $\endgroup$
    – Peter
    Oct 6, 2020 at 16:27

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How does one represent ...
There is this notation $$ a_0 + \underset{k=1}{\overset{\infty}{\sf K}} \frac{b_k}{a_k} = a_0 + \cfrac{b_1}{a_1 + \cfrac{b_2}{a_2 + \cfrac{b_3}{a_3 + \cfrac{b_4\;\;}{{\ddots}}}}}. $$

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