[seqfan] Re: A very fast convergent alternating series for phi

[mathar]

In response to http://list.seqfan.eu/pipermail/seqfan/2013-February/010846.html

vs> 
vs> (4)sum{k>=1}(-1)^(k-1)/(a(k)*a(k+1)) = 1/phi_N, where phi_N=(N+sqrt(N^2+4))/2 is a "metallic" ratio (for N=1-golden ratio, for N=2-silver ratio, etc.);  
vs> ...

one should point to the applicable literature on the subject of
summing inverse products of terms from the 2-term recurrences:

Gert Almkvist, A solution to a tantalizing problem, Fib. Quart. (1986) 316, 
http://www.fq.math.ca/Scanned/29-3/andre-jeannin1.pdf

Blagoj S. Popov, Summation of reciprocal series of numerical functions of second order, Fib. Quart. 24 (1) (1986) 17-21, http://www.fq.math.ca/Scanned/24-1/popov.pdf

R. Andre-Jeannin, Summation of certain reciprocal series related to Fibonacci and Lucas numbers, Fig. Quart. 29 (1991) 200, http://www.fq.math.ca/Scanned/29-3/andre-jeannin1.pdf

R. S. Melham, A generalization of a result of Andre-Jeannin concering summation of reciprocals, Proguliae Mathematica 57 (1) (2000) p. 45
http://www.emis.ams.org/journals/PM/57f1/pm57f104.pdf

F. Zhao, Notes on reciprocal series related to Fibonacci and Lucas Numbers, Fib. Quart. 39 (5) (2001) 392 , http://www.fq.math.ca/Scanned/39-5/zhao1.pdf

N. Omur, On reciprocal series of generalized Fibonacci numbers with subscripts in arithmetic progression, Discr. Dyn. Nat. Soc. 2012 #684280 http://dx.doi.org/10.1155/2012/684280

RJM