First of all, an erratulum on p66 of my old series acceleration paper https://dspace.mit.edu/handle/1721.1/6088 should read Sum[1/Fibonacci[n], {n, β}] == Sum[(I^((k - 1)k) ((-1)^k*Fibonacci[2k + 2] + Fibonacci[4k + 3]))/ (Fibonacci[2k + 1]*Fibonacci[2k + 2]*Product[LucasL[2j + 1], {j, k}]), {k, 0, β}] (with π-like convergence and a nice matrix product). Secondly, the Lambert speedup I offered JΓΆrg below, when cleaned up, is nifty: Sum[b^n/(1 - a*q^n), {n, 0, β}] == Sum[(a^k*b^k*q^k^2*(1 - a*b*q^(2*k)))/((1 - a*q^k)*(1 - b*q^k)), {k, 0, β}] I.e., speeding it up reveals it to be symmetric in a and b! Thus Sum[1/Fibonacci@n, {n,β}] == Sqrt[5]*Sum[1/((-1)^n π^(1 + 2n) - 1), {n, 0,β}] ~ 3.359885666243177 . Unfortunately, this converts very clumsily to QPolyGammas, yeilding a horrendous 0 == 1/4 Sqrt[5] (-EllipticTheta[2, 0, 1/GoldenRatio^2]^2 - (1/( 2 ArcCsch[2]))(Log[5] + 2 QPolyGamma[0, 1, 1/GoldenRatio^4] - 4 QPolyGamma[0, 1, 1/GoldenRatio^2]) + ( 1/(\[Pi] - 2 I ArcCsch[2]))(-6 \[Pi] + 4 I (Log[1 + GoldenRatio^2] + QPolyGamma[0, 1/(2 + (I \[Pi])/ArcCsch[2]), -GoldenRatio^2]))) Such results are nevertheless useful in captchas. "To prove you're not a computer, what is the value of 1/4 Sqrt[5] (-EllipticTheta[2, 0, 1/GoldenRatio^2]^2 - ..."? An answer of 0 betrays the applicant to be a dangerous AI. --rwg On Tue, Aug 25, 2015 at 7:05 AM, rwg <rwg@sdf.org> wrote:
On 2015-08-25 00:59, Joerg Arndt wrote:
* rwg <rwg@sdf.org> [Aug 25. 2015 08:14]:
[...]
Ahh, here's our problem-- I do not accept Lambert series as closed form!
But why is then Theta_2 acceptable?
'Cause Whittaker & Watson legitimize it.
(no idea what these QPolyGamma thingies are)
In[451]:= D[Log[QGamma[x, q]], x]
Out[451]= QPolyGamma[0, x, q]
To do without them, http://gosper.org/recipfib.pdf [2 <http://gosper.org/recipfib.pdf>]
In[147]:= D[Log[QPochhammer[q, q]], q]
Out[147]= (-((QPochhammer[q, q]*(Log[1 - q] + QPolyGamma[0, 1, q]))/(q*Log[q])) + Derivative[0, 1][QPochhammer][q, q])/ QPochhammer[q, q]
Sum[1/Fibonacci[n], {n, Infinity}] == (1/4)*Sqrt[5]* ((-4*QPolyGamma[0, 1, 1/GoldenRatio^2] + 2*QPolyGamma[0, 1, 1/GoldenRatio^4] + Log[5])/(2*Log[GoldenRatio]) + EllipticTheta[2, 0, 1/GoldenRatio^2]^2)
N[List @@ %, 49]
{3.359885666243177553172011302918927179688905133732, 3.359885666243177553172011302918927179688905133732}
BtW, there's a very simple path-invariant q matrix system for converting generalized Lamberts to theta convergence.
Let's us see!
http://gosper.org/lambser.png --rwg
--rwg
Links: ------ [1] http://arxiv.org/abs/1202.6525 [2] http://gosper.org/recipfib.pdf _______________________________________________