On 2015-11-09 16:29, rcs@xmission.com wrote:
Nice png, interesting paper. Initially, I just had the sides of the "triangles" approximated by line segments. The areas appeared hopelessly unequal. Amazing what a little bending can accomplish.
I like the old unslick proof that H(inf) = inf:
So does Master Zack.
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 + ...
gosper.org/zeta(2).pdf
which indeed has the property that it can be rearranged to sum to any real value -- perhaps pi, or zeta(3).
Gack. Obviously not with the kind of rearrangement in question. This "rearrangement" is just the usual evaluation by telescopy, not the sandwiching of the zeta 2 series.
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
I think you mean bounded
distance, which would cover this telescoping case.
It ought to be trivially obvious when that distance is 1. I've never seen (non "creative") telescopy where anyone worried about illegal rearrangement. --rwg
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... [2] 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.
Links: ------ [1] http://ma.sdf.org/gosper.org/zeta(2).pdf [2] http://mathcircle.berkeley.edu/archivedocs/2009_2010/lectures/0910lecturespd...