Math 2.0

Sequences, series, summability Series: basic notions Example

1 − 1 + 2 − 6 + 24 − 120 + ⋯

Builds on Borel summation

Example

The alternating series of factorials is ∑k=0∞(−1)kk!=1−1+2−6+24−120+⋯ .\sum_{k=0}^{\infty} (-1)^k k! = 1 - 1 + 2 - 6 + 24 - 120 + \cdots. Its terms do not tend to 00, since k!→∞k! \to \infty. So it is a divergent series.

Remarks

The series is Borel summable. Its Borel transform is ∑k≥0(−1)kk!k!tk=∑k≥0(−t)k=11+t\sum_{k \ge 0} \frac{(-1)^k k!}{k!} t^k = \sum_{k \ge 0} (-t)^k = \frac{1}{1 + t} for ∣t∣<1\abs{t} < 1. The function 1/(1+t)1/(1+t) is analytic on a neighbourhood of [0,∞)[0, \infty), and the Borel sum is ∫0∞e−t1+t dt≈0.596347.\int_0^\infty \frac{e^{-t}}{1 + t} \, dt \approx 0.596347. This number is known as the Gompertz constant. Euler studied this series and gave it the same value. The formal series ∑k(−1)kk! xk\sum_k (-1)^k k! \, x^k is the asymptotic expansion of ∫0∞e−t1+xt dt\int_0^\infty \frac{e^{-t}}{1 + x t} \, dt as x→0+x \to 0^+, and the example is this series taken formally at x=1x = 1.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

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Analytic continuation, concept, no evidence levelAnalytic continuationconceptFunction, concept, no evidence levelFunctionconceptSequence, definition, no evidence levelSequencedefinitionLaplace transform, definition, no evidence levelLaplace transformdefinitionReal sequence, definition, no evidence levelReal sequencedefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionSeries, definition, no evidence levelSeriesdefinitionPower series, definition, no evidence levelPower seriesdefinitionRadius of convergence, definition, no evidence levelRadius of convergencedefinitionBorel summation, definition, no evidence levelBorel summationdefinition1 − 1 + 2 − 6 + 24 − 120 + ⋯, example, no evidence level1 − 1 + 2 − 6 + 24 − …example
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Analytic continuation concept
  2. Function concept
  3. Sequence definition
  4. Laplace transform definition
  5. Real sequence definition
  6. Bounded sequence definition
  7. Series definition
  8. Power series definition
  9. Radius of convergence definition
  10. Borel summation definition
  11. 1 − 1 + 2 − 6 + 24 − 120 + ⋯ example

Further reading

  1. Web https://en.wikipedia.org/wiki/1_%E2%88%92_1_%2B_2_%E2%88%92_6_%2B_24_%E2%88%92_120_%2B_%E2%8B%AF
About this page
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MSC40G10, 40A05
Instance ofDivergent series
UsesBorel summation
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Changed2 November 2026

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