Crazy Summation

In order to solve this question, we are going to use a Borel summation.

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

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

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

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:

$\displaystyle \sum_{k=0}^\infty (-1)^{k} k! = \displaystyle \int_0^\infty \frac{\exp(-x)}{1+x} \, dx = e E_1 (1) \approx 0.5963\ldots$

where $E_1 (z)$ is the exponential integral.

Therefore, the answer is $0.596$