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In mathematics,

${\displaystyle \sum _{k=0}^{\infty }(-1)^{k}k!}$

is a divergent series, first considered by Euler, that sums the factorials of the natural numbers with alternating signs. Despite being divergent, it can be assigned a value of approximately 0.596347 by Borel summation.

Euler and Borel summation

This series was first considered by Euler, who applied summability methods to assign a finite value to the series. The series is a sum of factorials that are alternately added or subtracted. One way to assign a value to this divergent series is by using Borel summation, where one formally writes

${\displaystyle \sum _{k=0}^{\infty }(-1)^{k}k!=\sum _{k=0}^{\infty }(-1)^{k}\int _{0}^{\infty }x^{k}e^{-x}\,dx.}$

If summation and integration are interchanged (ignoring that neither side converges), one obtains:

${\displaystyle \sum _{k=0}^{\infty }(-1)^{k}k!=\int _{0}^{\infty }\lefte^{-x}\,dx.}$

The summation in the square brackets converges when ${\displaystyle |x|<1}$, and for those values equals ${\displaystyle {\tfrac {1}{1+x}}}$. The analytic continuation of ${\displaystyle {\tfrac {1}{1+x}}}$ to all positive real ${\displaystyle x}$ leads to a convergent integral for the summation:

${\displaystyle {\begin{aligned}\sum _{k=0}^{\infty }(-1)^{k}k!&=\int _{0}^{\infty }{\frac {e^{-x}}{1+x}}\,dx\\&=eE_{1}(1)\approx 0.596\,347\,362\,323\,194\,074\,341\,078\,499\,369\ldots \end{aligned}}}$

where E₁(z) is the exponential integral. This is by definition the Borel sum of the series, and is equal to the Gompertz constant.

Connection to differential equations

Consider the coupled system of differential equations

${\displaystyle {\dot {x}}(t)=x(t)-y(t),\qquad {\dot {y}}(t)=-y(t)^{2}}$

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

${\displaystyle x(t)=\sum _{n=1}^{\infty }(-1)^{n+1}{\frac {(n-1)!}{t^{n}}}}$

Observe x(1) is precisely Euler's series.

On the other hand, the system of differential equations has a solution

${\displaystyle x(t)=e^{t}\int _{t}^{\infty }{\frac {e^{-u}}{u}}\,du.}$

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 ${\displaystyle t=1}$, giving

${\displaystyle \sum _{n=1}^{\infty }(-1)^{n+1}(n-1)!=e\int _{1}^{\infty }{\frac {e^{-u}}{u}}\,du.}$

See also

• Alternating factorial
• 1 + 1 + 1 + 1 + ⋯
• 1 − 1 + 1 − 1 + ⋯ (Grandi's)
• 1 + 2 + 3 + 4 + ⋯
• 1 + 2 + 4 + 8 + ⋯
• 1 − 2 + 3 − 4 + ⋯
• 1 − 2 + 4 − 8 + ⋯

References

1. Euler, L. (1760). "De seriebus divergentibus" [On divergent series]. Novi Commentarii Academiae Scientiarum Petropolitanae (5): 205–237. arXiv:1808.02841. Bibcode:2018arXiv180802841E.

Further reading

• Kline, Morris (November 1983), "Euler and Infinite Series", Mathematics Magazine, 56 (5): 307–313, CiteSeerX 10.1.1.639.6923, doi:10.2307/2690371, JSTOR 2690371
• Kozlov, V. V. (2007), "Euler and mathematical methods in mechanics" (PDF), Russian Mathematical Surveys, 62 (4): 639–661, Bibcode:2007RuMaS..62..639K, doi:10.1070/rm2007v062n04abeh004427, S2CID 250892576
• Leah, P. J.; Barbeau, E. J. (May 1976), "Euler's 1760 paper on divergent series", Historia Mathematica, 3 (2): 141–160, doi:10.1016/0315-0860(76)90030-6

v t e Sequences and series
Integer sequences	Basic Arithmetic progression Geometric progression Harmonic progression Square number Cubic number Factorial Powers of two Powers of three Powers of 10 Advanced (list) Complete sequence Fibonacci sequence Figurate number Heptagonal number Hexagonal number Lucas number Pell number Pentagonal number Polygonal number Triangular number array
Properties of sequences	Cauchy sequence Monotonic function Periodic sequence
Properties of series	Series Alternating Convergent Divergent Telescoping Convergence Absolute Conditional Uniform
Explicit series	Convergent 1/2 − 1/4 + 1/8 − 1/16 + ⋯ 1/2 + 1/4 + 1/8 + 1/16 + ⋯ 1/4 + 1/16 + 1/64 + 1/256 + ⋯ 1 + 1/2 + 1/3 + ... (Riemann zeta function) Divergent 1 + 1 + 1 + 1 + ⋯ 1 − 1 + 1 − 1 + ⋯ (Grandi's series) 1 + 2 + 3 + 4 + ⋯ 1 − 2 + 3 − 4 + ⋯ 1 + 2 + 4 + 8 + ⋯ 1 − 2 + 4 − 8 + ⋯ Infinite arithmetic series 1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials) 1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series) 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)
Kinds of series	Taylor series Power series Formal power series Laurent series Puiseux series Dirichlet series Trigonometric series Fourier series Generating series
Hypergeometric series	Generalized hypergeometric series Hypergeometric function of a matrix argument Lauricella hypergeometric series Modular hypergeometric series Riemann's differential equation Theta hypergeometric series
Category

• Divergent series