* Bill Gosper <billgosper@gmail.com> [Sep 12. 2017 08:38]:
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
[...]
Uhm, see https://arxiv.org/abs/1202.6525 for this and other forms. Also the references, especially Thomas J.\ Osler, Abdul Hassen: On generalizations of Lambert's series, International Journal of Pure and Applied Mathematics, vol.43, no.4, pp.465-484, (2008). http://www.ijpam.eu/contents/2008-43-4/index.html Best regards, jj