numerical summation of series, rational case.

Michel Van den Bergh <michel.vandenbergh <at> uhasselt.be> writes:

> 
> Hi,
> 
> Is what you are doing comparing with series like
> 1=sum(1/n^k-1/(n+1)^k,n,1,inf)=k*sum(1/n^(k+1)+...,n,1,inf)?

  The code was a quick response, so it is not optimized, see the post to Fateman.


  b_n = 1/( n * (n+1) * (n+2) ... (n+k)) , sum(b_n,n,1,inf) is known by an easy
telescopic (there is a formula 1/(k*k!), for example:

  (7*n^3 + .../8*n^7+...) ---> principal = 7/8 n^(3-7) = 7/8 n^4 

 so I consider a_n - 7/8 * 1/n(n+1)(n+2)(n+3) which converge more quickly, 

 and sum a_n = sum 7/8 * 1/n(n+1)(n+2)(n+3) (known) + r_n , etc.

  Generalized factorial and its properties is well known.

  Local time is 11:30, time to go to bed  -:)


 Best regards,


 -M