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%I #50 Jul 09 2015 05:46:07
%S 5,1,5,4,1,8,4,5,5,8,2,5,4,1,7,9,9,1,3,3,0,1,1,9,1,9,9,6,3,9,4,2,9,8,
%T 7,1,1,0,4,5,6,7,9,1,8,5,4,8,0,1,5,8,5,2,9,1,7,3,6,7,1,8,6,6,0,9,8,7,
%U 9,1,8,9,7,2,1,4,9,9,0,5,7,0,1,3,2,0,5
%N Decimal expansion of Sum_{n>=1} (-1)^floor(n*sqrt(2))/n.
%C From _Jon E. Schoenfield_, Jul 08 2015: (Start)
%C If we define the partial sum s_n = Sum_{i=1..n} (-1)^floor(i*sqrt(2))/i then the real-valued sequence s_1, s_2, s_3, ... converges very slowly, and the convergence is not smooth because of the aperiodicity created by the (-1)^floor(i*sqrt(2)) factor. However, if we define the partial sum S_j = s_(n_j) where n_j is the j-th Pell number, then the real-valued sequence S_1, S_2, S_3, ... converges fairly quickly. It appears that, for either even or odd values of j, as j increases, S_j approaches
%C c0 + (c1 + k1*log(n))/n_j + (c2 + k2*log(n))/(n_j)^2 + (c3 + k3*log(n))/(n_j)^3 + ...,
%C but the coefficients c1, k1, c2, k2, c3, k3, etc. take one set of values when j is even and a different set of values when j is odd.
%C However, it also appears that, if we define the sequence of real numbers t_1, t_2, ... that results from adjusting each of the terms of the S_j sequence using
%C t_j = S_j - Sum{i=1..D} (-1)^(ij)*sqrt(2)*(j*d_i+(j mod 2)) / r^(j*(2i-1))
%C where r is the limit of the ratio Pell(j+1)/Pell(j) as j increases, i.e., r = 1 + sqrt(2), and d is the (empirically-determined) integer sequence
%C d = {0, 1, 3, 61, 395, 47041, 504987, 182501677, 2705354787, 2186736573121, ...}
%C (from which we use the first D terms in the above formula for t_j), then the sequence of real numbers t_1, t_2, ... follows the simpler form
%C c0 + C1/n_j + C2/(n_j)^2 + C3/(n_j)^3 + ...
%C where the coefficients c0, C1, C2, C3, etc. do not depend on the parity of j, and simple numerical methods can be used to evaluate those coefficients. E.g., if we use D=10 (or more), and if each of the values t_j is computed with a little more than 100 digits of precision, then c0 = -0.51541845582541799... can be obtained to 100 digits of precision by applying simple numerical methods to accelerate the convergence of the 21 values t_1, t_2, ..., t_21. (End)
%D A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 5: "Series", New York, Gordon and Breach Science Publishers, 1986-1992, p. 652, formula 6.
%H Jon E. Schoenfield, <a href="/A228639/b228639.txt">Table of n, a(n) for n = 0..100</a>
%H R. J. Mathar, <a href="/A228639/a228639.pdf">Approximations to sum_{n>=1} (-1)^floor(n*sqrt 2)/n</a>
%F Sum_{n>=1} (-1)^floor(n*sqrt(2))/n.
%e -0.51541845... (in reference only -0.5154, next digits computed by _Vaclav Kotesovec_).
%e -0.51541845582541799133011919963942987110456791854801... - _Jon E. Schoenfield_, Jul 08 2015
%Y Cf. A001951, A006337, A184774.
%K nonn,cons
%O 0,1
%A _Vaclav Kotesovec_, Aug 31 2013
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