H(2n) - H(n) -> log2 = .693... for large N. The H difference can be thought of as the area of a staircase, with steps of width 1/n and height n/k from the x-axis, as k goes from n+1 to 2n. [The height is from a fixed baseline; the drop from one step to the next is much smaller.] The staircase is a good approximation to the curve 1/x, 1<x<2, whose integral is log2. Rich -----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of wouter meeussen Sent: Wednesday, July 21, 2010 3:22 PM To: math-fun Subject: [math-fun] Nicole Oresme in the year 1350 and Zeta[1] remember the days when Nicholas O. showed elegantly that 1+1/2+1/3+1/4+1/5+... is unbounded? he stated, according to http://en.wikipedia.org/wiki/Nicole_Oresme that 1+1/2+(1/3+1/4)+(1/5+1/6+1/7+1/8)+ ... must be greater than 1+1/2+(1/4+/4)+(1/8+1/8+1/8+1/8)+... and so must be greater than 1+1/2+1/2+1/2+... a nice tidbit given by eric's http://mathworld.wolfram.com/HarmonicSeries.html My surprise: good old Mathematica 4.0 gives no solace for each of those cool brackets: Limit[Sum[1./k,{k,2^n+1,2^(n+1)}], n->Infinity] aka limit of HarmonicNumber[2^(n+1)]-HarmonicNumber[2^n] aka limit of -PolyGamma[0, 1 + 2^n] + PolyGamma[0, 1 + 2^(1 + n)] but, numerically, it looks a lot like Log[2]. The question "why?" is sometimes translated as "proof?" Wouter. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun