Notes on a General Sequence

Let {r_n}_n∈𝕅 be a strictly increasing sequence of nonnegative real numbers satisfying the asymptotic formula r_n ~ 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.