I posted the same question on Math Stackexchange: http://math.stackexchange.com/quest...g-harmonic-sums-with-residues/1085248#1085248 The answer there using complex analysis is great. I had questions, which Id like to get advice on here. (1) How did he get the laurent series at negative integers? I can never understand how that works. Doesnt he need to find the coefficient a_n? Which is defined as a contour integral? How does the Laurent series work for digamma? (2) How did he get the partial sum involving the q−1 in the upper index? And finally, I dont understand the step from: "which yields.. this implies that.." How did he make the transformation? It didnt change anything?