On 2014-07-07 00:39, Bill Gosper wrote:
[Adding subject; fixing bug.]
Whoa, how are you getting these? Convergence is absent or useless.
Holy crapoli, I completely missed the point of this Flanelle business. (Wasn't he the guy who helped Dumbledore make a Philosopher's Stone?)
WDS>
h(n) = (-1)^(parity of bit-sum of binary representation of n)
f(x) S=SUM[ f(n) * h(n), for n=0..2^k-1 with k large] ln(x+1) S=-log(2)/2 ln(x+2) S = -0.1379330125 ln(x+3) -0.07070756527 x^j 0 for any fixed integer j>=0 1/(x+1) 0.398761088108
[http://isc.carma.newcastle.edu.au/advancedCalc identifies this as 3^(1/6)/Zeta[3]^(5/237)/3 . I'm surprised you didn't notice.] http://arewomenhuman.me/wp-content/uploads/2012/10/trolls.jpeg
1/(x+2) 0.1049709156499 sqrt(x) -0.63407426 1/sqrt(x+1) 0.1983140804979
It appears that the cases f(x):=1/(x+k) (various k) are simply related. If S0:=SUM[h(n)/n, for n=1..2^k-1 with k large] ~ -1.19628326432525643722229, S1:=SUM[h(n)/(n+1), for n=0..2^k-1 with k large] ~ 0.398761088108 S2:=SUM[h(n)/(n+2), for n=0..2^k-1 with k large] ~ 0.1049709156499 S3:=SUM[h(n)/(n+3), for n=0..2^k-1 with k large] ~ 0.0419241705 then S0 = -3 S1 and S3 = (S1 - 3 S2)/2 . --rwg