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Let $\eta_i$,$\zeta_i$ be irrational real numbers, not necessarily distinct. $\eta_0$ may also be zero. I have two questions. My first question:

Can we express $1$ as $\eta_0+\dfrac1{\eta_1+\frac1{\eta_2+\frac1\ddots}}$? If so, can we also determine $\eta_j$?

And my second question:

Can we express $1$ as $\eta_0+\dfrac{\zeta_0}{\eta_1+\frac{\zeta_1}{\eta_2+\frac{\zeta_2}\ddots}}$? If so, can we also determine $\eta_j,\zeta_j$?

I am not sure how to answer this because I do not know exactly what qualifies as 'expressing $1$ as...' Surely, as $\zeta_i \to0$ and $\eta_0\to1$, the second c.f. converges to $1$. But this is not exactly 'representing' $1$ as a c.f. in the same sense as actually finding what distinct values of $\eta_i,\zeta_i$, when evaluated in the c.f., are equal to $1$. This ambiguity makes my second question a bit weak. My first question, however, does not have this ambiguity, as it is a simple continued fraction. The problem then is that I have made no progress on it.

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  • $\begingroup$ @saulspatz Those simple continued fractions have integer elements, so that doesn't apply to this question. $\endgroup$ – Robert Israel Dec 8 '19 at 5:05
  • $\begingroup$ @RobertIsrael I misread the question. Thanks. $\endgroup$ – saulspatz Dec 8 '19 at 12:27
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If $$ r = a_0 + \dfrac{1}{a_1 + \dfrac{1}{a_2 + \dfrac{1}{a_3 + \ldots}}}$$ is a simple continued fraction, where $r$ is a positive irrational number and $a_i$ are positive integers, then

$$ 1 = a_0/r + \dfrac{1}{a_1 r + \dfrac{1}{a_2/r + \dfrac{1}{a_3 r + \ldots}}}$$

expresses $1$ as a continued fraction with irrational elements $$\eta_i = \cases{a_i/r & if $i$ is even\\ a_i r & if $i$ is odd}$$

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