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Antilimit

From Wikipedia, the free encyclopedia

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

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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

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References

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  • 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.