# Antilimit

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In [mathematics](/source/mathematics), the '''antilimit''' is the equivalent of a [limit](/source/Limit_(mathematics)) for a [divergent series](/source/divergent_series). The concept is 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](/source/radius_of_convergence).

== Common divergent series ==

{| class="wikitable"
! Series !! Antilimit
|-
| [1 + 1 + 1 + 1 + ⋯](/source/1_%2B_1_%2B_1_%2B_1_%2B_%E2%8B%AF) || -1/2
|-
| [1 − 1 + 1 − 1 + ⋯ (Grandi's series)](/source/Grandi's_series) || 1/2
|-
| [1 + 2 + 3 + 4 + ⋯](/source/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF) || -1/12
|-
| [1 − 2 + 3 − 4 + ⋯](/source/1_%E2%88%92_2_%2B_3_%E2%88%92_4_%2B_%E2%8B%AF) || 1/4
|-
| [1 − 1 + 2 − 6 + 24 − 120 + …](/source/1_%E2%88%92_1_%2B_2_%E2%88%92_6_%2B_24_%E2%88%92_120_%2B_%E2%80%A6) || 0.59634736...
|-
| [1 + 2 + 4 + 8 + ⋯](/source/1_%2B_2_%2B_4_%2B_8_%2B_%E2%8B%AF) || -1
|-
| [1 − 2 + 4 − 8 + ⋯](/source/1_%E2%88%92_2_%2B_4_%E2%88%92_8_%2B_%E2%8B%AF) || 1/3
|-
| [1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series)](/source/Harmonic_series_(mathematics)) || <math> \gamma </math>
|}

== See also ==
* [Abel summation](/source/Abel_summation)
* [Cesàro summation](/source/Ces%C3%A0ro_summation)
* [Lindelöf summation](/source/Lindel%C3%B6f_summation)
* [Euler summation](/source/Euler_summation)
* [Borel summation](/source/Borel_summation)
* [Mittag-Leffler summation](/source/Mittag-Leffler_summation)
* [Lambert summation](/source/Lambert_summation)
* [Euler–Boole summation](/source/Euler%E2%80%93Boole_summation) and [Van Wijngaarden transformation](/source/Van_Wijngaarden_transformation) can also be used on divergent series

== References ==
*{{cite journal |last1=Shanks |first1=Daniel |title=An Analogy Between Transients and Mathematical Sequences and Some Nonlinear Sequence-to-Sequence Transforms Suggested by It. Part 1. |journal=Naval Ordnance Lab White Oak Md. |date=1949|url=https://apps.dtic.mil/sti/pdfs/ADA800123.pdf}}
*{{cite book |last1=Sidi |first1=Avram |title=Practical Extrapolation Methods |date=February 2010 |publisher=Cambridge University Press |isbn=9780511546815 |page=542|doi=10.1017/CBO9780511546815}}

{{Series (mathematics)}}

Category:Divergent series
Category:Summability methods
Category:Sequences and series
Category:Mathematical analysis

{{mathanalysis-stub}}

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Adapted from the Wikipedia article [Antilimit](https://en.wikipedia.org/wiki/Antilimit) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Antilimit?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.