Notes on a General Sequence

Abstract

Abstract Let {rn}n∈𝕅 be a strictly increasing sequence of nonnegative real numbers satisfying the asymptotic formula rn ~ αβn, where α, β are real numbers with α > 0 and β > 1. In this note we prove some limits that connect this sequence to the number e. We also establish some asymptotic formulae and limits for the counting function of this sequence. All of the results are applied to some well-known sequences in mathematics.