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After looking for explanations of the Euler's continued fractions to compute exponential on internet and after reading Euler's explanations about, I still don't understand how Euler find this continued fraction :

$$e=2+\dfrac{1}{1+\dfrac{1}{2+\dfrac{1}{1+\dfrac{1}{1+\dfrac{1}{4+\dfrac{1}{1+\dfrac{1}{1+\dfrac{1}{6+\ddots}}}}}}}}$$ I understand how Euler get continued fractions to compute squares but not for exponential. Maybe I have missed something, but I really need to understand. So thanks for your help.

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(slightly) related: Continued fraction for $\frac{1}{e-2}$ – Grigory M Dec 19 '13 at 19:23

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