[math-fun] Yet another zeta 2 proof
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.
Unfortunately, the paper perpetuates the old, unslick proof that Harmonic#(oo) = oo.
I'll bite. Old I grant, but unslick? -- -- http://cube20.org/ -- [ <http://golly.sf.net/>Golly link suppressed; ask me why] --
On 2015-11-09 10:22, Tom Rokicki wrote:
Unfortunately, the paper perpetuates the old, unslick proof that Harmonic#(oo) = oo.
I'll bite. Old I grant, but unslick?
Clever. Does that proof have a date? I've not seen it before. On Mon, Nov 9, 2015 at 11:51 AM, rwg <rwg@sdf.org> wrote:
On 2015-11-09 10:22, Tom Rokicki wrote:
Unfortunately, the paper perpetuates the old, unslick proof that Harmonic#(oo) = oo.
I'll bite. Old I grant, but unslick?
http://web.williams.edu/Mathematics/lg5/harmonic.pdf --rwg
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-- -- http://cube20.org/ -- [ <http://golly.sf.net/>Golly link suppressed; ask me why] --
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
Rich, I have to admit that I don't understand why searching for A000003 also pulls up sequences like A000364. If there were explicit crossreferences between them I would understand it. Maybe there used to be, and some traces of those old links are lodged in the OEIS's deep memory! But it is brilliant of the OEIS to do this, because there ARE connections, via the paper D. Shanks, Generalized Euler and class numbers <https://oeis.org/A000003/a000003.pdf>, Math. Comp. 21 (1967), 689-694; 22 (1968), 699. which mentions all the sequences that were displayed in the reply that you got. Magic! I will have to ask Russ Cox how he did it. Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Mon, Nov 9, 2015 at 7:29 PM, <rcs@xmission.com> wrote:
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
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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...
I'm pretty sure the solution to the Basel problem in this link is the same proof that someone presented to the Berkeley Math Circle some time between 3 and 5 years ago. I don't remember who it was that presented the proof. —Dan
On Nov 9, 2015, at 10:16 AM, Bill Gosper <billgosper@gmail.com> wrote:
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.
participants (6)
-
Bill Gosper -
Dan Asimov -
Neil Sloane -
rcs＠xmission.com -
rwg -
Tom Rokicki