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
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