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(posted & answered, then closed in MSE)

I've been interested in the Basel problem and its famous solution $$ \sum_{n=1}^{\infty}{\frac{1}{n^2}} = \frac{\pi^2}{6}. $$

Recently I saw this video along with a comment (highlighted from the link) stating

Cool fact: Euler actually approximated the sum to 16 decimal places and GUESSED that it was pi^2/6 before rigorously proving it

I know that Euler (and other mathematicians, related post) had approximated the sum by transforming it into a sum that converged quicker: $$ \sum_{n=1}^{\infty}\frac{1}{n^2} = \sum_{n=1}^{\infty}\frac{1}{2^n n^2} + (\ln2)^2, $$ but I never knew of the second part nor could I find any source that supports this comment. The trusty source ethanyap8680 is likely mixing/making things up.

So, is it even possible for Euler, or anyone for that matter, to guess the sum to be $\frac{\pi^2}{6}$ from the approximation $1.6449340668482264$? The only way I could see it is if they assumed there was some connection to $\pi$ along with some constant, specifically $$ 1.6449340668482264 \approx c\pi^k. $$

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    $\begingroup$ Leonh. Euler, "De summis serierum reciprocarum." Commentariii Academiae Scientiarum Imperialis Petropolitanae, Vol. 7 for the years 1734 & 1735, St. Petersburg 1740, pp. 123-134. "Huius autem seriei summam nuper ostendi proxime esse 1,6449340668482264364, ex cuius numeri sextuplo, si extrahatur radix quadrata, reipsa prodit numerus 3,141592653589793238 exprimens circuli ..." $\endgroup$ Commented Mar 3 at 3:47
  • $\begingroup$ @njuffa Woah this is an excellent source. Is there an English translation of the full document? $\endgroup$ Commented Mar 3 at 4:15
  • $\begingroup$ I don't know, but quite possibly yes, as this is a famous result. I don't have time to search for one right now. I did come across this web publication that claims to explain Euler's process: plus.maths.org/content/infinite-series-surprises $\endgroup$ Commented Mar 3 at 4:19
  • $\begingroup$ @njuffa That's alright. That specific quote and source essentially answers my question. Thank you. $\endgroup$ Commented Mar 3 at 4:31
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    $\begingroup$ In a quick search, I found English translations of Euler's original publication (Eneström index: E41) by Jordan Bell, Ian Bruce, and Alexander Aycock. $\endgroup$ Commented Mar 3 at 19:03

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Yes, Euler found first a numerical approximation, using what is called now the Euler-Maclaurin formula, and other tools which permitted him to guess the result. Connection to $\pi$ was suggested by other series whose sum was already known. More details are explained in the survey paper:

J. Lagarias, Euler's constant: Euler's work and modern developments Bull AMS, 50 4 (2013) 527-628,

in section 2.4.

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