How is it shown that the definition of the Euler-Mascheroni constant is finite?

Question

Sorry if it's trivial - but I would really like to know. Also, how were the hand-calculated approximations derived? A link to Euler's original publication would be perfect.

Thanks

Answer 1

The proof of finiteness with the least machinery: call $$ H_n = 1 + \frac{1}{2} + \cdots + \frac{1}{n} \; . $$ The sequence $$ a_n = H_n - \log n $$ starts a little high and decreases. The sequence $$ b_n = H_n - \log (n+1) $$ starts a little low and increases. we always have $b_n < a_n.$ However, $a_n - b_n = \log \frac{n+1}{n}$ goes to zero, so the sequences crash together at some point.

If you are worried, we actually have $$ b_m < a_n $$ for all pairs $m,n.$

Answer 2

From The Euler archive - E043 page on De progressionibus harmonicis observationes:

Euler gives $\gamma$ to six decimal places (only $5$ are correct, I think he says) and gives a formal summation that $\gamma$ equals $\frac{1}{2}s_2 - \frac{1}{3} s_3 + \frac{1}{4} s4 - \frac{1}{5} s_5 + \ldots\;$ [...]

According to the records, it was presented to the St. Petersburg Academy on March 11, 1734.

The page has links to the original publication, as well as English and German translations.