[math-fun] Kepler's pizza
(my favorite math teacher embarrasser) is now a click at http://www.tweedledum.com/rwg/, per prodding by MLB. The .mfe is http://gosper.org/pizza.mfe (not https, as I mistakenly sent someones last nite). Caution: earlier versions contain a bogus definition of "mean anomaly". --rwg
I have a formula that specializes to (c90) sum((-2)^?\2\-powers\-in(k)/k^2,k,1,inf) = %pi^2/12 inf ==== 2-powers-in(k) 2 \ (- 2) %pi (d90) > ------------------- = ---- / 2 12 ==== k k = 1 where 2_powers_in(2008) = 3, e.g.. Is this old news? It seems that (c96) sum(2^?\2\-powers\-in(k)/k^2,k,1,inf) = %pi^2/4 inf ==== 2-powers-in(k) 2 \ 2 %pi (d96) > --------------- = ----, / 2 4 ==== k k = 1 suggesting loads of other things to try. --rwg
Bill Gosper wrote:
I have a formula that specializes to (c90) sum((-2)^?\2\-powers\-in(k)/k^2,k,1,inf) = %pi^2/12
inf ==== 2-powers-in(k) 2 \ (- 2) %pi (d90) > ------------------- = ---- / 2 12 ==== k k = 1
where 2_powers_in(2008) = 3, e.g.. Is this old news?
Well, if we put r = 2-powers-in(k) and sum over r first, we get sum {r=0..oo} (-2)^r 4^-r sum {k=1,3,..oo} 1/k^2 which is just the product of two sums, one of which is 1-1/2+1/4-1/8... = 1/2+1/8+... = 2/3 and the other of which is famously pi^2/8. So it's certainly straightforwardly equivalent to old news :-). -- g
participants (2)
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Gareth McCaughan -
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