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I am writing a research paper on the Euler-Mascheroni Constant when I mentioned this sum: $$\gamma=\sum^\infty_{k=1}\left(\frac{1}{k}-\ln\left(1+\frac{1}{k}\right)\right)$$ which was derived by Euler. I saw this in wikipedia but it doesn't reference Euler's paper. Do you know where Euler writes this identity?

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    $\begingroup$ possibly here (top of p.157), but of course it doesn't use that notation. Someone who reads latin can confirm. $\endgroup$ Commented Nov 28, 2022 at 19:00
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    $\begingroup$ For future questions of this kind, consider History of Math and Science Stack Exchange $\endgroup$
    – njuffa
    Commented Nov 29, 2022 at 5:28
  • $\begingroup$ This follows directly from the limit definition of $\gamma$ as $$\sum_{k=1}^n\ln\left(1+\frac1k\right)=\sum_{k=1}^n[\ln(k+1)-\ln k]=\ln(n+1)=\ln n+o(1)$$ $\endgroup$
    – TravorLZH
    Commented Dec 11, 2022 at 0:38

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As Matthew Towers already identified in a comment to the question, the relevant publication is:

Leonh. Euler, "De Progressionibus Harmonicis Observationes." Commentarii Academiae Scientiarum Imperialis Petropolitanae, Tomus VII ad Annos MDCCXXXIV & MDCCXXXV. Petropolis, Typis Academiae 1740, pp. 150-161. (Leonh. Euler, "Observations on Harmonic Progressions", Notices of the Imperial Academy of Sciences of St. Petersburg, Vol. 7 for the years 1734 and 1735. St. Petersburg, Printer of the Academy 1740). A scan of the original publication can be found at the Bavarian State Library. The relevant portion is in §11, at the top of page 157:

Quae series, cum sint convergentes, si proxime sum-
mentur prodibit $1+\frac{1}{2}+\frac{1}{3} ---\frac{1}{i} = l(i + 1) + 0,577218$
Si summa dicatur $s$, foret, ut supra fecimus, $ds = \frac {di}{i+1}$,
ideoque $s=l(i+1)+C$. Huius igitur quantitatis con-
stantis $c$ valorem deteximus, quippe est $C=0,577218$.

Note that Euler uses $l$ in his publications to denote the natural logarithm. My translation, with changes for modern typography:

These series, since they are converging, when they are summed up approximately, will result in $1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{i} = \ln (i+1) + 0.577218$. If the sum is called $s$, it would be, as we did above, $ds = \frac {di}{i+1}$, and therefore $s=\ln(i+1)+C$. We therefore discovered the value of this constant quantity $c$, which is $C=0.577218$.

In general, when one searches for the first occurrence of a particular identity in the scientific record, one should expect that the historical notation used is quite different from modern notational conventions and preferences, and that it can therefore be quite challenging to identify such first occurrences, especially if one goes back several centuries.

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