MaplePrimes - answers and comments on Question, Change of variable in a sum

Summation by parts

Robert Israel — Mon, 24 Nov 2008 01:28:14 Z

Unfortunately, Maple does not have good facilities for low-level manipulation of sums. The sumtools or SumTools package would seem to be good places for these (analogous to IntegrationTools), but these are dedicated to sophisticated methods for finding closed-form summations. Well, here's a first attempt at writing a procedure for this. By the way, your rewriting is not quite right: it would require S(0) = 0 (try it for N=1). The procedure below corrects that.

SumByParts:= proc(E::Sum(algebraic,name=range), S::algebraic)
   local M,q,v,x,a,b,k;
   if op(0,E) = 'sum' then return Parts(Sum(op(E)),S) end if; 
   if not typematch(E, 
       Sum(q::algebraic,x::name=a::algebraic..b::algebraic),'M')
     then error "Wrong type" end if;
   assign(M);
   if type(q,`*`) and member(S,[op(q)]) then v:= q/S
   else v:= normal(q/S) 
   end if;
   Sum((S - eval(S,x=x-1))*sum(eval(v,x=k),k=x..b) 
        + eval(S,x=a-1)*v,x=a..b)
   end proc;

For example:

> Q:= Sum((a(j+1)-a(j))*j,j=1..N);
  SumByParts(Q, j);

Sum(a(N+1)-a(j),j = 1 .. N)

hm ...

Axel Vogt — Tue, 25 Nov 2008 23:29:20 Z

That does not work for, for your example I get "Error, (in Sum) Usage: Sum(expr,name) or Sum(expr,name=a..b) or Sum(expr,name=RootOf(...))"

Sum

Robert Israel — Wed, 26 Nov 2008 07:49:30 Z

Hmm... Did you by any chance have the student package loaded, or some other package that defines Sum ?
Normally Sum is not a defined procedure, so that error message couldn't occur.

aha ...

Axel Vogt — Thu, 27 Nov 2008 00:16:27 Z

Thank you: yes, I have it loaded for handy use of 'changevar' and will try again tonight without that

Edited: yes, not loading that package makes it work for me as well.