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
