limsupn

Limsupn, typically written as limsup_{n→∞} a_n or limsup_n a_n, denotes the limit superior of a sequence a_n as the index n tends to infinity. It captures the largest accumulation behavior of the sequence, even if the sequence does not converge.

Definition and notation: For a real sequence a_n, define s_n = sup{ a_k : k ≥ n }. The sequence s_n is nonincreasing in n, and the limit lim_{n→∞} s_n exists in the extended real numbers. The value limsup_{n→∞} a_n is this limit: limsup_{n→∞} a_n = lim_{n→∞} sup_{k≥n} a_k. Equivalently, limsup can be described as the supremum of all subsequential limits of a_n, when these limits exist.

Key properties: If a_n converges to a real number L, then limsup_{n→∞} a_n = L. In general, limsup a_n is greater than or equal to liminf_{n→∞} a_n, and every subsequence has a subsequence converging to a limit between these two bounds. If the sequence is unbounded above, limsup is +∞; if unbounded below, it may be −∞ in the extended sense.

Examples: For a_n = (−1)^n, the limsup is 1 and the liminf is −1. For a_n = 1/n, limsup is 0. For a_n = n modulo 2, the limsup is 1.

Extensions: Limsup can be defined for sequences of sets via limsup A_n = ∩_{n≥1} ∪_{k≥n} A_k, and for random sequences in probability and analysis, limits of tail suprema provide asymptotic behavior.