Symbolic summation (was: levin versus ratsum)

>    Thanks for the reference.  It gives neccesary and sufficient conditions for a
> rational function over the integer to be telescopic. The paper is very easy to
> read, i think that even high school pupils can get the general ideas.
> 
>   Perhaps i can improve ratsum to test cases in which the symbolic summation
> gives a rational number, like in the paper.
> 
> Best regards.
> 
>  Just for fun, you consider sum_n 1/(3n+1)^2 (*) and then  sum_n
> (1/(3n+1)^2+1/(3n+2)^2), the first one is sum_ an and the last one is
> sum b_n with b_n = a_n + a_(n+1) so trivially sum bn = (2* sum a_n) - a_1. 
> 
> 
> 

Hi,

I think one should distinguish between numeric and symbolic summation.
To be frank I think the levin transform is superior for numeric
summation (it is what Maple uses internally).

But a version of ratsum that implements the above symbolic algorithm
would be very nice I think. As I said Maxima seems to be sorely lacking
in symbolic summation capabilities.

Regards,
Michel


PS. Are we sure that this is not covered by Zeilberger? I think not
since Zeilberger is for hypergeometric series, but one should check.