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The infinite fraction $$f(n)=n+\frac{n}{n+\frac{n}{n+...}}$$ can be simplified to $$f(n)=\frac{n+\sqrt{n^2-4n}}{2}$$ However, I wanted to know if the fraction $$1+\frac{2}{3+\frac{4}{5+...}}$$ can be simplified further to get an answer using algebra or is the answer a transcendental number? On solving this in Desmos(till 25), I got 1.54149408254. This is an image of the Desmos page.

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    $\begingroup$ AFAIK, this can't be solved using algebra. It is transcendental, I am 99.99% sure. $\endgroup$
    – user840532
    Commented Dec 8, 2020 at 8:32
  • $\begingroup$ @LeonhardEuler You solved it, about 250 years ago! See my answer. $\endgroup$ Commented Dec 8, 2020 at 9:57
  • $\begingroup$ @ParclyTaxel yes, I didn't tell because I wanted to see whether today's mathematicians can solve it or not :p $\endgroup$
    – user840532
    Commented Dec 8, 2020 at 10:11

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The Mathworld page on continued fraction constants gives the exact answer as $$\frac1{\sqrt e-1}$$ which is indeed transcendental.

The value was known to Euler himself. See page 14 of this translation.

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  • $\begingroup$ But what does 'K' mean as a notation in the Mathworld page and how did they derive the answer? $\endgroup$
    – sato
    Commented Dec 9, 2020 at 11:14
  • $\begingroup$ @AyaanMaan I think they just referred to Euler's paper and the OEIS sequence attached to that continued fraction. The K is just another odd notation for continued fractions. $\endgroup$ Commented Dec 9, 2020 at 11:14

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