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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
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