---
tag: V24RQP5492
kind: example
name: 1 − 1 + 2 − 6 + 24 − 120 + ⋯
msc: [40G10, 40A05]
instance_of: [0JWMJSTBSG]
uses: [RQYX6E42D0]
sources:
  - url: https://en.wikipedia.org/wiki/1_%E2%88%92_1_%2B_2_%E2%88%92_6_%2B_24_%E2%88%92_120_%2B_%E2%8B%AF
same_as:
  wikidata: Q3511165
  wikipedia: 1 − 1 + 2 − 6 + 24 − 120 + ⋯
---

## Example

The alternating series of factorials is
$$\sum_{k=0}^{\infty} (-1)^k k! = 1 - 1 + 2 - 6 + 24 - 120 + \cdots.$$
Its terms do not tend to $0$, since $k! \to \infty$.
So it is a [[divergent series|0JWMJSTBSG]].

## Remarks

The series is [[Borel summable|RQYX6E42D0]].
Its Borel transform is $\sum_{k \ge 0} \frac{(-1)^k k!}{k!} t^k = \sum_{k \ge 0} (-t)^k = \frac{1}{1 + t}$ for $\abs{t} < 1$.
The function $1/(1+t)$ is analytic on a neighbourhood of $[0, \infty)$, and the Borel sum is
$$\int_0^\infty \frac{e^{-t}}{1 + t} \, dt \approx 0.596347.$$
This number is known as the Gompertz constant.
Euler studied this series and gave it the same value.
The formal series $\sum_k (-1)^k k! \, x^k$ is the asymptotic expansion of $\int_0^\infty \frac{e^{-t}}{1 + x t} \, dt$ as $x \to 0^+$, and the example is this series taken formally at $x = 1$.

## Explanations

### L1 human

The terms of this series grow so fast that the partial sums swing wildly between large positive and large negative numbers.
Still, one can give it a meaningful value.
Divide each term by $k!$ to get the geometric series of $1/(1+t)$, then undo the division with the integral $\int_0^\infty e^{-t} t^k \, dt = k!$.
This gives about $0.5963$.