# How is it shown that the definition of the Euler-Mascheroni constant is finite? [closed]

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

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