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?
Where does Euler write the definition $\gamma=\sum^\infty_{k=1}\left(\frac1k-\ln\left(1+\frac1k\right)\right)$ of the Euler-Mascheroni Constant
euler-mascheroni-constantreference-request
Related Solutions
$x^{1/x}$ has a maximum at $x=e$ with e as euler‘s number. (can be shown by taking the derivative of the function)
Because the function is increasing before and decreasing after $x=e$, if x and y are two real numbers, which are bigger than 0 and $x<y$, than if x and y are smaller than e:
$$x^{1/x}<y^{1/y}$$
Raise it to the power of $xy$.
$$x^y<y^x$$
This holds true because x and y are positive numbers.
In your case the two numbers are smaller than e, so this proof applies.
It is known that for $n\geq 1$, $$ \log n! > \left( {n + \frac{1}{2}} \right)\log n - n + \log \sqrt {2\pi } + \frac{1}{{12n}} - \frac{1}{{360n^3 }}. $$ Thus \begin{align*} \sum\limits_{n = 1}^\infty {\frac{{\log n!}}{{n^3 }}} & > \sum\limits_{n = 1}^\infty {\frac{{\log n}}{{n^2 }}} + \frac{1}{2}\sum\limits_{n = 1}^\infty {\frac{{\log n}}{{n^3 }}} - \sum\limits_{n = 1}^\infty {\frac{1}{{n^2 }}} \\ &\quad + \log \sqrt {2\pi } \sum\limits_{n = 1}^\infty {\frac{1}{{n^3 }}} + \frac{1}{{12}}\sum\limits_{n = 1}^\infty {\frac{1}{{n^4 }}} - \frac{1}{{360}}\sum\limits_{n = 1}^\infty {\frac{1}{{n^6 }}} \\ & = - \zeta '(2) - \frac{1}{2}\zeta '(3) - \zeta (2) + \zeta (3)\log \sqrt {2\pi } + \frac{1}{{12}}\zeta (4) - \frac{1}{{360}}\zeta (6) \\ & = 0.583661 \ldots>\gamma. \end{align*} For the values of the derivative of $\zeta$ I used the OEIS A073002 and A244115. The other values, except $\zeta(3)$, can be expressed in terms of $\pi$.
Best Answer
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:
Note that Euler uses $l$ in his publications to denote the natural logarithm. My translation, with changes for modern typography:
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.