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Rate of Convergence of Positive Series

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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 = (λn), 0 = λ0 < λn ↑ + ∞, n → + ∞, β = {βn: 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 = (λn) 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; +∞).

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 12, pp. 1665 – 1674, December, 2004.

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Skaskiv, O.B. Rate of Convergence of Positive Series. Ukr Math J 56, 1975–1988 (2004). https://doi.org/10.1007/s11253-005-0162-2

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Keywords

  • Nonnegative Function
  • Positive Series