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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>