What Mathematica "says" on two BBP type infinite sums ?

*To*: mathgroup at smc.vnet.net*Subject*: [mg103765] What Mathematica "says" on two BBP type infinite sums ?*From*: Alexander Povolotsky <apovolot at gmail.com>*Date*: Mon, 5 Oct 2009 07:39:14 -0400 (EDT)

Hello, What Mathematica (I don't have it) "says" on two BBP type infinite sums ? 1) sum(((1/(exp(Pi)-log(3))/log(2))^n)/(n^3+2*n^2+2*n+7),n = 0 ... infinity) ISC *tells* that 95*sum(((1/(exp(Pi)-log(3))/log(2))^n)/(n^3+2*n^2+2*n+7),n = 0 ... infinity) + 8*(Pi)^2 - 146*Catalan + 20* Pi*log(2) -6*(log(2))^2 ~~ 0 Perhaps 95*sum(1/((204461223889343/13382366326351)^n)/(n^3+2*n^2+2*n+7),n = 0 .. infinity) is even more close to 8*(Pi)^2 - 146*Catalan + 20* Pi*log(2) -6*(log(2))^2 ? Any further improvements to (...)^n factor to get closer to 8*(Pi)^2 - 146*Catalan + 20* Pi*log(2) -6*(log(2))^2 ? 2) 2*sum(1/(n^3+2*n^2+2*n+7)/(24)^n,n = 0 .. infinity) ISC *tells* that this sum is close to Pi*sqrt(3) - 39*log(3) + 84*log(2) + 25*gamma -8*Pi*sqrt(2) It appears that this sum's proximity to Pi*sqrt(3) - 39*log(3) + 84*log(2) + 25*gamma -8*Pi*sqrt(2) could be improved if instead of 24 23.999999995011916301243901392414554490409136352246963766236377143509476137987495024510254888936149797331254797 be used Any further improvements (desirably in the close form ;-) ) ? Thanks, Cheers, Alexander R. Povolotsky

**Follow-Ups**:**Re: What Mathematica "says" on two BBP type infinite sums***From:*Daniel Lichtblau <danl@wolfram.com>