Antilimit
In mathematics, the antilimit is the equivalent of a limit for a divergent series. The concept not necessarily unique or well-defined, but the general idea is to find a formula for a series and then evaluate it outside its radius of convergence.
Common divergent series
| Series | Antilimit |
|---|---|
| 1 + 1 + 1 + 1 + ⋯ | -1/2 |
| 1 − 1 + 1 − 1 + ⋯ (Grandi's series) | 1/2 |
| 1 + 2 + 3 + 4 + ⋯ | -1/12 |
| 1 − 2 + 3 − 4 + ⋯ | 1/4 |
| 1 − 1 + 2 − 6 + 24 − 120 + … | 0.59634736... |
| 1 + 2 + 4 + 8 + ⋯ | -1 |
| 1 − 2 + 4 − 8 + ⋯ | 1/3 |
| 1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series) | - |
See also
- Abel summation
- Cesàro summation
- Lindelöf summation
- Euler summation
- Borel summation
- Mittag-Leffler summation
- Lambert summation
- Euler–Boole summation and Van Wijngaarden transformation can also be used on divergent series
References
- Shanks, Daniel (1949). "An Analogy Between Transients and Mathematical Sequences and Some Nonlinear Sequence-to-Sequence Transforms Suggested by It. Part 1" (PDF). Naval Ordnance Lab White Oak Md.
- Sidi, Avram (February 2010). Practical Extrapolation Methods. Cambridge University Press. p. 542. doi:10.1017/CBO9780511546815. ISBN 9780511546815.
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