On 2015-09-15 14:25, Veit Elser wrote:
On Sep 15, 2015, at 5:04 PM, Bill Gosper <billgosper@gmail.com> wrote:
Out[305]= 1 - n + n*Log[n] - Inactive[Sum][Log[z], {z, 2, n}] == (-(1/24))*Inactive[Sum][ ConditionalExpression[ (-1 + 3*¢*(1 + ¢)* (-1 + 10*¢*(1 + ¢)* (1 + 12*¢*(1 + ¢))) + 180*¢^3*(1 + ¢)^3* (1 + 2*¢)*(Log[¢] - Log[1 + ¢]))/(15*¢^3*(1 + ¢)^3), Re[¢] > 0 || Re[¢] < -1 || NotElement[¢, Reals]], {¢, 1, -1 + n}] + Inactive[Sum][((-1)^¢*BernoulliB[1 + ¢]* Derivative[¢][-Log[1 + #1] + Log[n + #1] & ][0])/ (1 + ¢)!, {¢, 0, 3}] I see you have gone beyond asserting “my two cents” by summing to ¢=3.
-Veit
At least I kept it real. In the 2¢ plane you can run into copyright problems. [big chop] The function is
EulMac[xp_, v_, lo_, hi_, deg_, y_: 0] := Integrate[xp /. v -> y + v, {v, lo, hi}] - Inactive[Sum][xp, {v, lo + 1, hi}] == (-1)^deg*Inactive[Sum][ Integrate[(deg + 1)*v^deg* D[xp /. v -> v*y + ¢ + 1, {v, deg + 1}] - BernoulliB[deg + 1, y + v]* D[xp /. v :> y + v + ¢, {v, deg + 1}], {v, 0, 1}], {¢, lo, hi - 1}]/(deg + 1)! + Inactive[Sum][(-1)^¢* Derivative[¢][Evaluate[(xp /. v -> # + hi) - (xp /. v -> # + lo)] &][y]* BernoulliB[¢ + 1, y]/(¢ + 1)!, {¢, 0, deg}]
The motivation for porting was NSum's seeming inability to get so much as four digits of Sum 2 to oo of
1/Log[n!] - 1/Log[(2 n)!] - 1/Log[(-1 + 2 n)!]
apparently due to NSum's internal Euler-Maclaurin failing to try Method -> "DoubleExponential" on the corresponding NIntegrate. I now believe the sum ~ 1.4424630245831935 .
It may be merely 1/ln2 ~ 1.44269504, depending on how gradually f(n) can decrease before Limit[-f(n+2)-f(n+3)-...-f(2n+1),n->∞] = Sum(n>0)(f(n+1) - f(2n) - f(2n+1)) is no longer 0. Can this limit vanish w/o (non-oscillatory) f converging? E.g., can we construct an f where Limit[-f(n+2)-f(n+3)-...-f(2n+1),n->∞] =0 but Limit[-f(n+2)-f(n+3)-...-f(3n+1),n->∞] >0 ? Empirically, the sum seems to vanish for f := 1/n/ln(n+a), even though Sum f diverges. Or maybe it's just very small: < 6e-15 for a=1. Can we lower this bound? For the problem at hand, f(n):=1/Log[(n+1)!], which converges only a little slower than 1/n/ln(n+1), but the smallest magnitude EulMac and NSum have found is -.0002, with a supposedly more careful result of -.00035. Can anyone help with these sums? --rwg
This is still not accurate enough for PSLQ [n]or the like, and ISC humorously proposes 1/Log[2 + 1/(300 Khinchin Gamma[1/6])]. --rwg