Nice png, interesting paper. I like the old unslick proof that H(inf) = inf: grouping terms is a natural technique, and the idea extends naturally to other series; it can also be used to prove upper bounds, such as the zeta(2) series. Note that the proffered proof that zeta(2) < 2, based on telescoping 1/(n2+n), needs extra justification, since it involves rearranging a non-absolutely-converging series: 1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + ... which indeed has the property that it can be rearranged to sum to any real value -- perhaps pi, or zeta(3). I assume there's some lemma that it's ok to rearrange a non-abs-conv series if the terms -> 0, and no term moves more than a finite distance, which would cover this telescoping case. The paper also mentions that the problem of summing 1/n2 goes back to at least 1644. Can anyone recommend a good math history for the 17th century? It would be interesting to trace the ideas that led to calculus and (old-style) analysis. Gregory's arctan series has always puzzled me. Rich PS: Neil, *I* like A3. But when I type it into the search bar, it returns A3 first, but also secant & tangent numbers, and more. I don't see any connection to class numbers. PPS: Do you have corresponding series for sqrt(positive) & cbrts? ---------- Quoting Bill Gosper <billgosper@gmail.com>:
Young eavesdropper Zack sent me http://mathcircle.berkeley.edu/archivedocs/2009_2010/lectures/0910lecturespd... a weird proof based on (X,Y) = (log(sin(x+y)/sin(x)), log(sin(x+y)/sin(y)) being (to me, remarkably) area-preserving. I think the paper needs a picture of this. gosper.org/passare3.png --rwg Unfortunately, the paper perpetuates the old, unslick proof that Harmonic#(oo) = oo. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun