1 − 1 + 2 − 6 + 24 − 120 + ...
In mathematics, the divergent series
was first considered by Euler, who applied summability methods to assign a finite value to the series.[1] 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 we formally write
If we interchange summation and integration (ignoring the fact that neither side converges), we obtain:
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:
where 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 time derivatives.
The solution with stable equilibrium at as has . And substituting it into the first equation gives us a formal series solution
Observe is precisely Euler's series.
On the other hand, we see the system of differential equations has a solution
By successively integrating by parts, we recover the formal power series as an asymptotic approximation to this expression for . Euler argues (more or less) that setting equals to equals gives us
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 + ⋯
- 1 − 1 + 1 − 1 + ⋯ (Grandi's)
- 1 + 2 + 3 + 4 + ⋯
- 1 + 2 + 4 + 8 + ⋯
- 1 − 2 + 3 − 4 + ⋯
- 1 − 2 + 4 − 8 + ⋯
References
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Further reading
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