On certain sequences derived from generalized Euler-Mascheroni constants

Abstract

Let 0 < α < 1 , and let Cα := lim n→∞ ( 1+ 1 2α + · · ·+ 1 nα − n 1−α 1−α ) . It is proved that there exists a unique sequence (ωn) such that 1+ 1 2α + · · ·+ 1 nα = Cα + (n+ωn)1−α 1−α . Moreover, the sequence (ωn) is decreasing and satisfies 2 ωn 1 4 [ 1+ ( 1+ n )α] , whence limn→∞ωn = 2 . This is only a special case of the more general results established in this paper. These results concern some sequences derived from generalized Euler–Mascheroni constants involving convex functions and complement similar ones obtained by V. Timofte [Integral estimates for convergent positive series. J. Math. Anal. Appl. 303 (2005), 90–102]. Mathematics subject classification (2010): 11B83, 26D15.