Nearalternating

Nearalternating is a descriptive term used in the study of sequences and series to indicate a sign pattern that closely follows, but does not strictly adhere to, alternation. In many contexts, a sequence is considered nearalternating if the signs of its nonzero terms agree with the alternating pattern (-1)^n except for a sparse set of indices where the sign deviates.

Formally, let a_n be a real sequence with infinitely many nonzero terms, and define s_n = sign(a_n) for those n with a_n ≠ 0. The sequence is nearalternating if s_n coincides with (-1)^n for all n outside a small or sparse set. Variants include the case where the set of deviations has finite size, or where its natural density within the first N indices tends to zero as N grows.

Examples include a sequence whose terms are a_n = (-1)^n/n for all n except at finitely many indices where the sign is flipped, producing a small, controlled departure from strict alternation. Such perturbations typically do not destroy convergence properties that hold for purely alternating sequences.

Properties and implications vary with context. If |a_n| decreases to zero and the sign deviations are finite or sufficiently sparse, many results that hold for alternating series may extend to nearalternating sequences, preserving convergence in cases where the underlying alternating series would converge. However, the exact impact depends on the magnitude and distribution of the deviations.

Nearalternating is related to concepts like almost alternating sequences and sign patterns in series analysis. See also: alternating sequence, almost alternating sequence.