Euler-Gompertz constant

The Euler-Gompertz constant is the value of the continued fraction

$$C_2=0+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{2}{1+\mathrm{\dots }}}}},$$

in which after three appearances of 1 in the numerator position, follow the integers from 2 forward each given twice consecutively; the value of this constant is approximately 0.596347362323194074341078499369279376074… Finch gives two formulas for this constant:

$$C_2=-e\text{Ei}(-1)={\int }_1^{\mathrm{\infty }}\frac{\text{exp}(1-x)}{x}𝑑x,$$

with $e$ being the natural log base and Ei being the exponential integral.

The constant can also be expressed as a formula involving an infinite sum:

$$e\left(\left(\sum _{i=1}^{\mathrm{\infty }}\frac{{(-1)}^{i-1}}{i!i}\right)-\gamma \right),$$

with $\gamma$ being the Euler-Mascheroni constant.