ON THE SUM AND MAXIMAL TERM OF TAYLOR-DIRICHLET TYPE SERIES - Bodnarchuk, A. Yu.; Skaskiv, O. B. (PRECARPATHIAN BULLETIN OF THE SHEVCHENKO SCIENTIFIC SOCIETY. Number, 2024)
The article deals with Taylor-Dirichlet type series of the form F(x)=olimits_k=0^+∞a_ke^xλ_k+τ(x)β_k, where (λ_k) and (β_k) are some sequence of non-negative numbers, and τ(x) is a non-negative non-decreasing function, a_k≥ 0 (k≥ 0). The class of such functions we denote TD(Λ,β,τ). The main statement of the paper is Theorem 2: Let a sequence (λ_n+β_n) be increasing, a sequence β=(β_n) be non-decreasing and a positive function τ be such that τ(x+h)-τ(x)h (x > 0, h>0). If the condition _k=0^∞(λ_k+1+β_k+1-λ_k-β_k)^-1 <+∞ is fulfilled, then for every function FTD(Λ,β,τ) the asymptotic relation F(x)=(1+o(1))μ(x,F) holds as x→ +∞ outside some set E[0,+∞) of finite Lebesgue measure (_Edx<+∞), where μ(x,F)={|a_k|e^τ(x)β_k+xλ_kk≥ 0}. Theorem 2 was proved earlier (1998) under the conditions of strict increasing of the sequences (λ_n) and (β_n).