I don't know the relation or even the integrals, just thought the lengthy Wolfram article would help -- it mentions some integrals, perhaps not the ones you meant. Sorry the reference didn't help.
Sent from my iPad On May 19, 2012, at 12:27 AM, "R.E. Boss" <r.e.b...@planet.nl> wrote: > 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.htmlyuy >> >> 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 ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
