I know where to find the Bernoulli numbers, but I'm asking for the relation with the integrals to be calculated.
R.E. Boss > -----Oorspronkelijk bericht----- > Van: programming-boun...@jsoftware.com > [mailto:programming-boun...@jsoftware.com] Namens km > Verzonden: zaterdag 19 mei 2012 1:04 > Aan: Programming forum > Onderwerp: Re: [Jprogramming] Challenge 12(?) > > See http://mathworld.wolfram.com/BernoulliNumber.html > > Sent from my iPad > > > On May 18, 2012, at 4:33 PM, "R.E. Boss" <r.e.b...@planet.nl> wrote: > > > 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 > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
