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 we formally write

$\sum_{k=0}^\infty (-1)^k k! = \sum_{k=0}^\infty (-1)^k \int_0^\infty x^k \exp(-x) \, dx$

If we interchange summation and integration (ignoring the fact that neither side converges), we obtain:

$\sum_{k=0}^\infty (-1)^k k! = \int_0^\infty \left[\sum_{k=0}^\infty (-x)^k \right]\exp(-x) \, dx$

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, we obtain a convergent integral for the summation:

$\sum_{k=0}^\infty (-1)^{k} k! = \int_0^\infty \frac{\exp(-x)}{1+x} \, dx = e E_1 (1) \approx 0.596347362323194074341078499369\ldots$

where $E_1 (z)$ 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 time derivatives.

The solution with stable equilibrium at $(x,y)=(0,0)$ as $t\to\infty$ has $y(t)=1/t$. And substituting it into the first equation gives us a formal series solution

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

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

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

$x(t) = e^{t}\int^{\infty}_{t}\frac{e^{-u}}{u}\mathrm{d}u.$

By successively integrating by parts, we recover the formal power series as an asymptotic approximation to this expression for $x(t)$. Euler argues (more or less) that setting equals to equals gives us

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

Results

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

k	Increment calculation	Increment	Result
0	1 · 0! = 1 · 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", Novi Commentarii academiae scientiarum Petropolitanae (5): 205, arXiv:1202.1506

Further reading

• Kline, Morris (November 1983), "Euler and Infinite Series", Mathematics Magazine 56 (5): 307–313, doi:10.2307/2690371
• 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