The Euler-Mascheroni Constant is defined as :

$$ {\displaystyle {\begin{aligned}\gamma =\lim _{n\to \infty }\left(-\ln n+\sum _{k=1}^{n}{\frac {1}{k}}\right)\\[5px]=\int _{1}^{\infty }\left(-{\frac {1}{x}}+{\frac {1}{\lfloor x\rfloor }}\right)\,dx.\end{aligned}}} $$

and equates to approximately 0.5772.

The constant was first described in 1734, and has now been (computationally) calculated to trillions of digits. It is still not known whether the number is 'rational'. In other words whether it's an infinite sequence or not.

Further information Wolfram MathWorld