On Sun, Mar 4, 2012 at 6:45 PM, Bill Gosper <billgosper@gmail.com> wrote:
in http://gosper.org/fst.pdf, namely
Sum[((-1)^n* Cos[Sqrt[(n + 1/2)^2 + a^2]*Pi])/((n + 1/2)*(n - a + 1/2)*(n + a + 1/2)), {n, 0, Infinity}] == (Pi*(Sin[a*Pi]^2 - 2*Sin[(a*Pi)/Sqrt[2]]^2))/(2*a^2*Cos[a*Pi])
gives oo=oo for a = 1/2+ (integer k), but gives
(6*(-1)^k*Cos[((1 + 2*k)*Pi)/Sqrt[2]])/(1 + 2*k)^3 + ((-1)^k*Sqrt[2]*Pi*Sin[((1 + 2*k)*Pi)/Sqrt[2]])/(1 + 2*k)^2
if you skip the n=k term.
Correction: n = |k+1/2| - 1/2
--rwg With help from Corey and Julian, and buffoonery from Mathematica.
Identity (5) generalizes to the bilateral sum (op. cit. last page) Sum[(((-1)^n*Cos[Sqrt[(phi + n + a/b)*(phi + n + a*b)]*Pi])/(phi + n))*(phi + n - a), {n, -Infinity, Infinity}] == (Pi*((Sin[a*Pi]*Cos[(phi - a)*Pi])/Sin[phi*Pi] - 2*Sin[(a*(b + 1)*Pi)/(2*Sqrt[b])]^2))/(a*Sin[(phi - a)*Pi]) which blows up when phi approaches any integer k, unless the sum skips n=-k, whereupon both sides equal (4*b*Cos[a*Pi] - 2*a*Pi*(2*b*Cos[(a*(1 + b)*Pi)/Sqrt[b]]*Csc[a*Pi] + (1 + b^2)*Sin[a*Pi]))/(4*a^2*b) (independent of k). Similarly, when phi approaches k+a, the sum must skip n=-k to give the finite value (2*Cos[(a*(1 + b)*Pi)/Sqrt[b]] + a*Pi*(-2*Cot[a*Pi] + ((1 + b)*Sin[(a*(1 + b)*Pi)/Sqrt[b]])/Sqrt[b]))/(2*a^2) Identity (5) came from specializing parameters and then "folding the sum in half". This somehow makes it sensitive to which term blows up. --rwg
Consider the standard binary tree with infinitely many levels.
Suppose each edge is colored green with probability = p.
What is the probability f(p) that there exists an infinite green path starting at the root?
--Dan
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f(p) = (2p - 1)/p^2 for p > 1/2, f(p) = 0 for p <= 1/2.
-- Gene
Yow, slope 8 at p=1/2+ ! Maybe building a tree is a good way to test coins for fairness. --rwg
mrob>There is also an intentional surcharge for oddly-shaped packages, for the packing reason you mentioned and for other reasons. rwg>Shouldn't I get a discount for shipping twice as many regular tetrahedra as regular octahedra of the same edge length? --rwg