# 1 − 1 2 − 6 24 − 120 ...

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In mathematics, the divergent series

## Contents

k = 0 ( 1 ) k k !

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 alternatingly are added or subtracted. A way to assign a value to the divergent series is by using Borel summation, where one formally writes

k = 0 ( 1 ) k k ! = k = 0 ( 1 ) k 0 x k e x d x

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

k = 0 ( 1 ) k k ! = 0 [ k = 0 ( x ) k ] e x d x

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:

k = 0 ( 1 ) k k ! = 0 e x 1 + x d x = e E 1 ( 1 ) 0.596 347 362 323 194 074 341 078 499 369

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

x ˙ ( t ) = x ( t ) y ( t ) , 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 ) = n = 1 ( 1 ) n + 1 ( n 1 ) ! t n

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

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

x ( t ) = e t t e u u d u .

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

n = 1 ( 1 ) n + 1 ( n 1 ) ! = e 1 e u u d u .

## Results

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

## References

1 − 1 + 2 − 6 + 24 − 120 + ... Wikipedia

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