In mathematics, the divergent series

## Contents

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

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

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:

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

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

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

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

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

## Results

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