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Say you have the infinitely continued fraction: $$x+\cfrac{1}{x^2+\cfrac{1}{x^3+\cfrac{1}{x^4+\ddots}}}$$

When $x=1$, you can see that it's $$1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\ddots}}}$$ which converges upon the golden ratio $\phi = 1.61803398875...$.

What if I were to plug in $x=2$? $$2+\cfrac{1}{4+\cfrac{1}{8+\cfrac{1}{16+\ddots}}}$$ Using a caluclator, it seems to converge upon the irrational number: $2.24248109286...$

Is there any way I can represent this irrational number algebraically?

metamorphy
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Sam
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  • The chances are that you can't represent this number algebraically. The chances are that it is transcendental, like $\pi$, not algebraic, like $\root3\of2$. – Gerry Myerson Aug 15 '15 at 00:38
  • @GerryMyerson Are you sure? – Sam Aug 15 '15 at 00:40
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    It has very good rational approximations. The theorem of Thue-Siegel-Roth says algebraic irrationals can't have rational approximations that are too good. That's where I'd recommend starting, if you want to try to prove the number is transcendental. – Gerry Myerson Aug 15 '15 at 00:46

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Let $F(x)=x^0+\cfrac1{x^1+\cfrac1\cdots}~.$ Then $F(2)$ is OEIS A$214070$, for which no closed form is currently known.

Lucian
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  • I haven studied these much but this question made me want to look into them. I'm wondering if there is a nice series representation of this continued fraction – David P Aug 15 '15 at 07:04
  • There is a little more information at https://oeis.org/A096641 – Gerry Myerson Aug 15 '15 at 12:12