Antilimit

{{No inline|date=May 2025}} In mathematics, the '''antilimit''' is the equivalent of a limit for a 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.

== Common divergent series ==

{| class="wikitable" ! 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) || <math> \gamma </math> |}

== 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 == *{{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}}

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