# Euler-Gompertz constant

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

 $C_{2}=0+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{2}{1+\ldots}}}},$

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:

 $C_{2}=-e\textrm{Ei}(-1)=\int_{1}^{\infty}\frac{\textrm{exp}(1-x)}{x}dx,$

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:

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

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

## References

• 1 Steven R. Finch, Mathematical Constants. Cambridge: Cambridge University Press (2003): 424
Title Euler-Gompertz constant EulerGompertzConstant 2013-03-22 18:49:06 2013-03-22 18:49:06 PrimeFan (13766) PrimeFan (13766) 7 PrimeFan (13766) Definition msc 11A55 Gompertz constant