Absorption (logic)

In mathematics, the divergent series

$\sum _{k=0}^{\mathrm{\infty }}(-1{)}^{k}k!$

was first considered by Euler, who applied resummation methods to assign a finite value to the series. The series is a sum of factorials that alternatingly are added or subtracted. A simple way to sum the divergent series is by using Borel summation:

$\sum _{k=0}^{\mathrm{\infty }}(-1{)}^{k}k!=\sum _{k=0}^{\mathrm{\infty }}(-1{)}^k\underset{0}{\overset{\mathrm{\infty }}{\int }}{x}^k\exp (-x)dx$

If we interchange summation and integration, we obtain:

$\sum _{k=0}^{\mathrm{\infty }}(-1{)}^{k}k!=\underset{0}{\overset{\mathrm{\infty }}{\int }}[\sum _{k=0}^{\mathrm{\infty }}(-x{)}^k]\exp (-x)dx$

The summation in the square brackets converges and equals 1/(1 + x) if x < 1. If we replace the summation by 1/(1 + x) regardless of whether it converges, we obtain a convergent integral for the summation:

$\sum _{k=0}^{\mathrm{\infty }}(-1{)}^{k}k!=\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{\exp (-x)}{1+x}dx=e{E}_1(1)\approx 0.596347362323194074341078499369\dots$

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

Results

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

k	Increment calculation	Increment	Result
0	1 · 0! = 1 · 1	1	1
1	−1 · 1	−1	0
2	1 · 2 · 1	2	2
3	−1 · 3 · 2 · 1	−6	−4
4	1 · 4 · 3 · 2 · 1	24	20
5	−1 · 5 · 4 · 3 · 2 · 1	−120	−100
6	1 · 6 · 5 · 4 · 3 · 2 · 1	720	620
7	−1 · 7 · 6 · 5 · 4 · 3 · 2 · 1	−5040	−4420
8	1 · 8 · 7 · 6 · 5 · 4 · 3 · 2 · 1	40320	35900
9	−1 · 9 · 8 · 7 · 6 · 5 · 4 · 3 · 2 · 1	−362880	−326980

See also

• 1 + 1 + 1 + 1 + · · ·
• Grandi's series
• 1 + 2 + 3 + 4 + · · ·
• 1 + 2 + 4 + 8 + · · ·
• 1 − 2 + 3 − 4 + · · ·
• 1 − 2 + 4 − 8 + · · ·

References

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