1 − 1 + 2 − 6 + 24 − 120 + ...

In mathematics, the divergent series

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

was first considered by Euler, who applied summability methods to assign a finite value to the series.^[1] The series is a sum of factorials that alternatingly are added or subtracted. A way to assign a value to the 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:

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

The summation in the square brackets converges and equals 1/1 + x if x < 1. If we analytically continue this 1/1 + x for all real x, one obtains a convergent integral for the summation:

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

where E₁(z) is the exponential integral. This is by definition the Borel sum of the series.

Derivation

Consider the coupled system of differential equations

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

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

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

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 setting equals to equals gives

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

Results

The results for the first 10 values of k are shown below:

k	Increment calculation	Increment	Result
0	1	1	1
1	−1 × 1	−1	0
2	1 × 2 × 1	2	2
3	−1 × 3 × 2 × 1	−6	−4
4	1 × 4 × 3 × 2 × 1	24	20
5	−1 × 5 × 4 × 3 × 2 × 1	−120	−100
6	1 × 6 × 5 × 4 × 3 × 2 × 1	720	620
7	−1 × 7 × 6 × 5 × 4 × 3 × 2 × 1	−5040	−4420
8	1 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1	40320	35900
9	−1 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1	−362880	−326980

See also

• 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:1202.1506.

Further reading

• Kline, Morris (November 1983), "Euler and Infinite Series", Mathematics Magazine, 56 (5): 307–313, doi:10.2307/2690371, JSTOR 2690371
• Kozlov, V. V. (2007), "Euler and mathematical methods in mechanics" (PDF), Russian Mathematical Surveys, 62 (4): 639–661, doi:10.1070/rm2007v062n04abeh004427
• 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