eng Science and Education Publishing Turkish Journal of Analysis and Number Theory 2014-09-08 2 4 147 151 10.12691/tjant-2-4-8 TJANT2014248 article Mansour Mahmoud mansour@mans.edu.eg 1 Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia In the paper, the author shows that the partial sums are alternatively larger and smaller than the generalized Euler's harmonic numbers with sharp bounds, where is the Euler's constant, are the Bernoulli numbers and is the digamma function. http://pubs.sciepub.com/tjant/2/4/8/tjant-2-4-8.pdf Euler constant ψ-function harmonic numbers inequalities asymptotic expansion sharp bounds