How can I find the exact value of this continued fraction?

Asked Aug 06 '21 at 12:34

Active Aug 06 '21 at 12:51

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How can I find the value of $$1 + \cfrac{1}{1 + \cfrac{1}{2 + \cfrac{1}{4 + \cfrac{1}{8 + \cfrac{1}{16 + \cfrac{1}{32 + \cfrac{1}{64 + \ldots}}}}}}}?$$ I have tried approximating the continued fraction and the result is close to $1.6915$. How can I find the exact value of this continued fraction?

edited Aug 06 '21 at 12:51

soupless

2,344

asked Aug 06 '21 at 12:34

İsmim Yok

1

These are called continued fractions not series. – Gary Aug 06 '21 at 12:39
2

@Community \cfrac should be used for continued fractions, not \dfrac. – soupless Aug 06 '21 at 12:50
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Ok @soupless, Thank you, will remember – UNAN Aug 06 '21 at 12:53
See OEIS A096641, they reference Ramanujan and the Regular Continued Fraction Expansion of Real Numbers. – Vepir Aug 06 '21 at 13:09
Try https://www.wolframalpha.com/input/?i=1%2BN%5BContinuedFractionK%5B2%5En%2C+%7Bn%2C+0%2C+400%7D%5D%2C+1000%5D – Claude Leibovici Aug 06 '21 at 13:46
For the fun of it $\frac{1602-542 e+978 e^2 } {1290+247 e+323 e^2 }$ gives 20 exact significant figures – Claude Leibovici Aug 06 '21 at 13:48

How can I find the exact value of this continued fraction?

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