Euler's constant

Euler's constant (sometimes called the Euler–Mascheroni constant) is a mathematical constant, usually denoted by the lowercase Greek letter gamma (γ), defined as the limiting difference between the harmonic series and the natural logarithm, denoted here by log: γ = lim n → ∞ ( ∑ k = 1 n 1 k − log ⁡ n ) = ∫ 1 ∞ ( 1 ⌊ x ⌋ − 1 x ) d x . {\displaystyle {\begin{aligned}\gamma &=\lim _{n\to \infty }\left(\sum _{k=1}^{n}{\frac {1}{k}}-\log n\right)\\&=\int _{1}^{\infty }\left({\frac {1}{\lfloor x\rfloor }}-{\frac {1}{x}}\right)\,\mathrm {d} x.\end{aligned}}} Here, ⌊·⌋ represents the floor function. The numerical value of Euler's constant, to 50 decimal places, is: