Rahul Sharma (Editor)

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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