In mathematics, is a divergent series, first considered by Euler, that sums the factorials of the natural numbers with alternating signs.
[1] The series is a sum of factorials that are alternately added or subtracted.
This is by definition the Borel sum of the series, and is equal to the Gompertz constant.
Consider the coupled system of differential equations where dots denote derivatives with respect to t. The solution with stable equilibrium at (x,y) = (0,0) as t → ∞ has y(t) = 1/t, and substituting it into the first equation gives a formal series solution Observe x(1) is precisely Euler's series.
On the other hand, the system of differential equations has a solution By successively integrating by parts, the formal power series is recovered as an asymptotic approximation to this expression for x(t).
Euler argues (more or less) that since the formal series and the integral both describe the same solution to the differential equations, they should equal each other at