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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)	${\displaystyle \gamma }$

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.

v t e Sequences and series
Integer sequences	Basic Arithmetic progression Geometric progression Harmonic progression Square number Cubic number Factorial Powers of two Powers of three Powers of 10 Advanced (list) Complete sequence Fibonacci sequence Figurate number Heptagonal number Hexagonal number Lucas number Pell number Pentagonal number Polygonal number Triangular number array
Properties of sequences	Cauchy sequence Monotonic function Periodic sequence
Properties of series	Series Alternating Convergent Divergent Telescoping Convergence Absolute Conditional Uniform
Explicit series	Convergent 1/2 − 1/4 + 1/8 − 1/16 + ⋯ 1/2 + 1/4 + 1/8 + 1/16 + ⋯ 1/4 + 1/16 + 1/64 + 1/256 + ⋯ 1 + 1/2 + 1/3 + ... (Riemann zeta function) Divergent 1 + 1 + 1 + 1 + ⋯ 1 − 1 + 1 − 1 + ⋯ (Grandi's series) 1 + 2 + 3 + 4 + ⋯ 1 − 2 + 3 − 4 + ⋯ 1 + 2 + 4 + 8 + ⋯ 1 − 2 + 4 − 8 + ⋯ Infinite arithmetic series 1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials) 1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series) 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)
Kinds of series	Taylor series Power series Formal power series Laurent series Puiseux series Dirichlet series Trigonometric series Fourier series Generating series
Hypergeometric series	Generalized hypergeometric series Hypergeometric function of a matrix argument Lauricella hypergeometric series Modular hypergeometric series Riemann's differential equation Theta hypergeometric series
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