[Jprogramming] Challenge 12(?)

[Viktor Cerovski]

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. at -,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
> 
> 

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