Zbl 0762.05009 Gosper, William Strip mining in the abandoned orefields of nineteenth century mathematics. (English) [CA] Computers and mathematics, Proc. Int. Conf., Stanford/CA (USA) 1986, Lect. Notes Pure Appl. Math. 125, 261-284 (1990).

Hi, Sergey, the reviewer blew it. The correct formula is \sum {n\ge 1}{2^{-n}\over 1+x\^{2^{-n}}}={1 \over \log x}+{1 \over 1-x}. E.g., (c195) sum(2^-n/(1+x^2^-n),n,1,inf) = 1/log(x)+1/(1-x) inf ==== \ 1 1 1 (d195) > ------------ = ------ + ----- / 1 log(x) 1 - x ==== -- n = 1 n n 2 2 (x + 1) (c196) apply_nouns(subst([x = 0.5d0,inf = 64],%)) (d196) 0.55730495911104d0 = 0.55730495911104d0 This is really not deep. You can change it to a finite, telescoping sum of k terms by replacing x by x^2^-k and subtracting. I wonder if we can get the review corrected. Actually, I'm happy it got reviewed at all (English?), since that paper contains an important idea that the world is ignoring. --Bill Gosper PS, I uploaded the Strip-Mining TeX sources and dvis to http://gosper.org/ stanfordn1.tex, ... , stanfordn4.tex You might also enjoy the identities at www.tweedledum.com/rwg/idents.htm