Segments of exponential series and regularly varying sequences

Abstract

The task of this paper is to investigate asymptotic behavior of segments of exponential series defined as $$T_{\lambda }(x):=\sum _{n<\lambda x}{\displaystyle \frac{c_n}{n!}}{x}^n,\lambda ın{R}^{+},x\to ınfty,$$where $(c_n{)}_{nınN}$ belongs to the set of regularly varying sequences in Karamata sense of arbitrary index. Precise results are obtained.