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I learned about continued fraction, and I wondered whether this form of fraction can be a complex number. $$1-\cfrac1{1-\cfrac1{1-\cfrac1{1-\cfrac1\ddots}}}$$

This cannot be convergent, but I can't prove it. Is there any way to prove the non-convergence of it?

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  • $\begingroup$ Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. $\endgroup$
    – Community Bot
    Commented May 21 at 14:34
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    $\begingroup$ The continued fraction is $(1-1,1,-1,1,-1,...)$, so the sequence of convergents oscillates between $1,0,\infty$, and thus it is not convergent. $\endgroup$ Commented May 21 at 15:10
  • $\begingroup$ It is divergent indeed. en.wikipedia.org/wiki/… $\endgroup$ Commented May 21 at 15:30
  • $\begingroup$ for periodic continued fractions with minus signs, see Zagier's reduction of indefinite binary quadratic forms. Here is something, does not directly do the fractions though math.hawaii.edu/~kmanguba/mastersproject.pdf $\endgroup$
    – Will Jagy
    Commented May 21 at 18:18
  • $\begingroup$ As explained in the link above to Wikipedia, $1+ \frac{z\mid}{\mid1}+\frac{z\mid}{\mid1}+\frac{z\mid}{\mid1}+\dots$ diverges for $z\in(-\infty,-1/4)$. But please avoid "no clue" questions: if you 'just want to know how to show this fraction does not converge and does oscillate', why not edit your post to include your calculation of the sequence of convergents? (see Zhang's comment above). $\endgroup$ Commented May 22 at 7:41

1 Answer 1

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$$a=1-\frac{1}{a}$$ $$a^{2}=a-1$$ $$a^{2}-a+1=0$$ Using quadratic equation: $$a=0.5\pm \frac{i\sqrt{3}}{2}$$ Hence I think this converges?

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  • $\begingroup$ Not really, I think it doesn't converge. But I just want to know how to show this fraction does not converge and does oscillate. $\endgroup$ Commented May 21 at 15:49

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