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?

  • $\begingroup$ The chances are that you can't represent this number algebraically. The chances are that it is transcendental, like $\pi$, not algebraic, like $\root3\of2$. $\endgroup$ – Gerry Myerson Aug 15 '15 at 0:38
  • $\begingroup$ @GerryMyerson Are you sure? $\endgroup$ – Sam Aug 15 '15 at 0:40
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    $\begingroup$ 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. $\endgroup$ – Gerry Myerson Aug 15 '15 at 0:46

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.

  • $\begingroup$ 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 $\endgroup$ – David P Aug 15 '15 at 7:04
  • $\begingroup$ There is a little more information at oeis.org/A096641 $\endgroup$ – Gerry Myerson Aug 15 '15 at 12:12

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