rLimitteja

rLimitteja is a concept in the field of sequence transformation and summation methods. It refers to a family of limit values obtained from a real sequence by applying a parameterized smoothing process. The idea generalizes classical summation techniques such as Cesàro and Abel summation, providing a way to study the limit behavior of sequences that may not have an ordinary limit.

Formally, for a real sequence (a_n), define A_n(r) = (1 - r) ∑_{k=0}^{n} r^k a_k for 0 < r < 1. If the limit L(r) = lim_{n→∞} A_n(r) exists, then L(r) is called an rLimitteja of the sequence. The mapping r ↦ L(r) yields a family of limit values that describe how the sequence behaves under r-weighted averaging. The parameter r controls the degree of smoothing: smaller r gives more weight to early terms, while larger r gives more weight to later terms.

Several basic properties follow. If a_n converges to a finite value L, then L(r) = L for every r in (0,1). For certain divergent sequences, L(r) may exist for some r even when the ordinary limit does not; the function r ↦ L(r) then characterizes the sequence’s smoothed limits. For example, if a_n = (-1)^n, then L(r) = (1 - r)/(1 + r). The concept is connected to Abel summation and serves as a pedagogical tool in numerical analysis and theoretical discussions of limit methods.

See also: Cesàro sum, Abel summation, sequence transformation.