Rate of Convergence of Positive Series

Abstract

We investigate the rate of convergence of series of the form

$$F(x) = \mathop \sum \limits_{n = 0}^{ + \infty } \;a_n e^{x\lambda _n + \tau (x)\beta _n } ,\quad a_n \geqslant 0,\quad n \geqslant 1,\quad a_0 = 1$$

where λ = (λ_n), 0 = λ₀ < λ_n ↑ + ∞, n → + ∞, β = {β_n: n ≥ 0} ⊂ ℝ₊, and τ(x) is a nonnegative function nondecreasing on [0; +∞), and

$$F(x) = \mathop \sum \limits_{n = 0}^{ + \infty } \;a_n f(x\lambda _n ),\quad a_n \geqslant 0,\quad n \geqslant 1,\quad a_0 = 1,$$

where the sequence λ = (λ_n) is the same as above and f (x) is a function decreasing on [0; +∞) and such that f (0) = 1 and the function ln f(x) is convex on [0; +∞).