What is the relation between the integrals ( http://www.jsoftware.com/jwiki/Pi/AConvergentSeries ) and the Bernoulli numbers? Or where can I find it?
R.E. Boss > -----Oorspronkelijk bericht----- > Van: programming-boun...@jsoftware.com > [mailto:programming-boun...@jsoftware.com] Namens Viktor Cerovski > Verzonden: dinsdag 15 mei 2012 18:54 > Aan: programming@jsoftware.com > Onderwerp: Re: [Jprogramming] Challenge 12(?) > > > > R.E. Boss wrote: > > > > I scanned a part of Exploratory Experimentation and Computation, > > https://www.opendrive.com/files?57384074_sCfMP , where two > > statements are made about a very rapidly convergent series. > > > > 1. The first term coincides with (pi % 8) in the first 42 digits > > 2. The first 2 terms even give 500 digits > > > > How can these two statements be confirmed (or rejected) with J? > > > They can be confirmed by calculating the integrals. > > Bernoulli numbers are involved among other things, and Roger's essay > http://www.jsoftware.com/jwiki/Essays/Bernoulli%20Numbers > gives > > B0=: 3 : 0 > b=. ,1x > for_m. }.i.x: y do. b=. b,(1+m)%~-+/b*(i.m)!1+m end. > ) > > which can be shortened to: > > B0t=: [: ([ , +/@:(* i.!>:) -@% 1+])~/ }:@i.@-,1: > > B0 20 > 1 _1r2 1r6 0 _1r30 0 1r42 0 _1r30 0 5r66 0 _691r2730 0 7r6 0 _3617r510 0 > 43867r798 0 > > B0t 20x > 1 _1r2 1r6 0 _1r30 0 1r42 0 _1r30 0 5r66 0 _691r2730 0 7r6 0 _3617r510 0 > 43867r798 0 > > ts 'y0=.B0 200' > 1.85257 362496 > > ts 'y0t=.B0t 200x' > 1.8853 293888 > > y0-:y0t > 1 > > > > > > > > > (No deadlines apply.) > > > > > > R.E. Boss > > > > ---------------------------------------------------------------------- > > For information about J forums see http://www.jsoftware.com/forums.htm > > > > > > -- > View this message in context: > http://old.nabble.com/Challenge-12%28-%29-tp33844136s24193p33849223.html > Sent from the J Programming mailing list archive at Nabble.com. > > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
