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Re: error with Sum and Infinity

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  • Subject: [mg102479] Re: error with Sum and Infinity
  • From: Richard Fateman <fateman at>
  • Date: Tue, 11 Aug 2009 04:02:06 -0400 (EDT)
  • References: <h5ol0j$tr$>

Bill Rowe wrote:
> On 8/9/09 at 6:19 PM, fateman at (Richard Fateman)
> wrote:
>> You and I know that Mathematica didn't do the right thing. But there
>> was no way for a user to determine if the problem was ill-posed in
>> terms of Mathematica.
> I disagree. The problem posed was an infinite sum of terms all 0
> except at for one term with a value of 1. Any real thought about
> the posed problem should come to the clear conclusion
> Mathematica cannot literally sum an infinite series.

So far, I agree entirely.

> For an
> arbitrary series, it would seem the apparent the most direct way
> to evaluate such a sum would be to sample various terms. Such an
> approach with the specific series should lead one to expect the
> result 0. It seems to me for problems of this sort, there is a
> lack of thinking about what is known and drawing reasonable
> conclusions from that.

I still agree entirely.

Where I differ is that there is a distinct possibility that a program 
could look at the definition of the summand and notice that it was a 
constant everywhere in the interval except at one place, which leads 
immediately to a method to do the summation.  This is what you propose 
that a human would do, and I suggest that a program could do the same 
thing.  Indeed, given the cleverness of programmers at WRI, I would not 
know whether or not Mathematica would get the right or wrong answer here 
without trying it.  Furthermore, none of us can tell whether some future 
version of Mathematica might get this right, as stated.  If the summand 
violated some stated rule of construction and was not well-formed, then 
we could say that it is unexpected that Mathematica would get the right 
answer today or any time in the future.

This general issue is, by the way, not unique to Mathematica, but occurs 
in other systems in which expressions and programs are interchanged. 
For reasons of evaluation ordering, and also human nature, this problem 
seems to crop up in summands and in the arguments of plotting commands.

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