R.E. Boss 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? >
The first integral has the form: integral cos(2x) * product_n cos(x/n) dx When you write the product of cos(x/n) as exp of sum of ln cos(x/n), series expansion of ln cos(x/n) has Bernoulli numbers in its coefficients. Obtained double-sum series then should be transformed a bit and appropriately truncated so that it has sufficient precision, and then numerical integration performed. The second expression (expansion of pi/8) is written in terms of integrals with different cosine terms and the same infinite product term, so truncation and numerical integration has to be done with even higher precision for m=1, etc. In short, there is quite a bit of work to get this through in J. > > 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 > > -- View this message in context: http://old.nabble.com/Challenge-12%28-%29-tp33844136s24193p33877568.html Sent from the J Programming mailing list archive at Nabble.com. ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
