Re: Re: error with Sum and Infinity

*To*: mathgroup at smc.vnet.net*Subject*: [mg102340] Re: [mg102318] Re: error with Sum and Infinity*From*: "Elton Kurt TeKolste" <tekolste at fastmail.us>*Date*: Thu, 6 Aug 2009 06:30:12 -0400 (EDT)*References*: <200908050944.FAA18234@smc.vnet.net>

Having skimmmed the responses I find the distinction between the actual numerical sum and the ability of Mathematica to manipulate a symbolic expression to be appropriate. Apparently Mathematica has a well-defined transition point from rote calculation to symbolic manipulation: In[144]:= t[i_] := 0; t[3] = 1; {Sum[t[i], {i, 1000000}], Sum[t[i], {i, 1000001}]} Out[144]= {1, 0} On Wed, 05 Aug 2009 05:44 -0400, "Bill Rowe" <readnews at sbcglobal.net> wrote: > On 8/4/09 at 4:30 AM, fateman at cs.berkeley.edu (Richard Fateman) > wrote: > > >The underlying point is that Mathematica is conflating two concepts > >with the name Sum: > > >A. A loop of finitely many terms evaluated in sequence and adding > >up the terms. and > > >B. A symbolic calculation based on various combinatorial ideas, the > >calculus of finite differences, and other systematic simplifications > >that reduces a summation, either finite or infinite, into a result > >that does not have any summation notation in it. Like summing > >arithmetic progressions, geometric progressions, etc (and very > >advanced etc.). > > >For this second concept to work, the summand must be something that > >can be suitably manipulated, typically starting as a single > >algebraic expression. A programming segment, or a pattern match that > >requires that each value of the index be fed into an evaluator will > >not, generally work with algorithms for indefinite or > >definite/infinite summation. Obviously you cannot feed an infinite > >number of index values into a function and sum up all the terms. > > >A clean solution would be to separate these two concepts: a loop > >and a symbolic closed-form simplifier for a summation. Or for > >Mathematica to use the Sum form, but somehow allow you to indicate > >to the system that you want it evaluated as a loop or simplified to > >a closed form. > > Your suggestion truly will not solve anything. > > Suppose Mathematica had functions SumA and SumB per your > description above. Now instead of a new user asking why Sum > doesn't return 1 when infinitely many terms exist, you will get > questions as to why SumB cannot deal with a finite number of > terms or the same question when SumA fails to work as the user > is expecting. > > It does not help clarify things to add a new function with a > different name for new users. In fact, it is more likely to > cause confusion since you would now have two things with similar > names that don't behave the same. And if the two names were not > similar, you almost certainly ensure a new user will not find > the one with the less intuitive name. > > And for the more experienced/knowledgeable user there is no > advantage to what you propose over the way things are now. > > Regards, Kurt Tekolste

**References**:**Re: error with Sum and Infinity***From:*Bill Rowe <readnews@sbcglobal.net>