I am asking about the original way that Euler did to calculate Napier's constant, I heard that he was able to compute its first 23 decimals.

  • $\begingroup$ I'm not sure how Euler did it, but a naive approach such as Maclaurin's series seems to converge reasonably quickly (there are faster methods such as Brothers' formula, but that's more modern). Still, 23 decimal places seems extremely tedious using the tools of his day: do you have a reference to support this? $\endgroup$ – Deepak May 23 '17 at 0:10
  • $\begingroup$ @Deepak I actually read it in a website here is the link link $\endgroup$ – Hks.Adohkd May 23 '17 at 8:22

The article Napier's $e$ - $e$ in Leonhard Euler's Introductio says that Euler used the series $$ e=1+{1\over 1}+{1^2\over 1\cdot 2}+{1^3\over 1\cdot 2\cdot 3}+\cdots $$ to obtain $$ e \approx 2.71828182845904523536028\cdots $$

Wikipedia cites Euler's original text, which confirms the story:

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  • $\begingroup$ To get an accuracy of $23$ d.p. (as per the OP), Euler would've had to compute the reciprocal of factorials up to $24!$, presumably by hand. That seems... unlikely. $\endgroup$ – Deepak May 23 '17 at 0:40
  • $\begingroup$ I suppose it's possible with repeated long division for an extremely determined and patient man haha. $\endgroup$ – Deepak May 23 '17 at 0:45
  • $\begingroup$ @Deepak, you're underestimating Euler's calculation skills and determination! $\endgroup$ – lhf May 23 '17 at 3:08
  • $\begingroup$ LOL. But (apart from being a genius) was he a savant in calculation? $\endgroup$ – Deepak May 23 '17 at 3:46
  • $\begingroup$ I'm really confused now, @lhf cited Euler's original text but on the other hand I'm with Deepak , I don't think that he was able to compute 23 d.p. from this formula $\endgroup$ – Hks.Adohkd May 23 '17 at 9:07

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