sum command...

[webmasterpdx]

I'm not a CAS expert, so you could look at the XCas source code if you really want to see how it does it, but I do know if you try the sum() function with arguments that are functions of different variables, it'll be a lot more complex than simply calling the same function over and over again. e.g. This SUM(1/y^2,y,x,1000x) results in Psi(x,1)-Psi(1000*x+1,1).

The sigma function is way way faster as it's just doing what that pseudocode does (calling the same function over and over again). e.g. Sum with an upper limit of 5000 takes about 20 seconds, but sigma returns an answer immediately.

FYI, the XCas source code is here...
https://www-fourier.ujf-grenoble.fr/~parisse/giac/src/