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Re: Re: error with Sum and Infinity

  • To: mathgroup at smc.vnet.net
  • Subject: [mg102307] Re: [mg102301] Re: error with Sum and Infinity
  • From: Daniel Lichtblau <danl at wolfram.com>
  • Date: Wed, 5 Aug 2009 05:42:40 -0400 (EDT)
  • References: <h56bq2$buv$1@smc.vnet.net> <200908040830.EAA26993@smc.vnet.net>

Richard Fateman wrote:
> You've gotten several of the standard work-arounds or excuses, which are 
> really  standard issue here.
> 
> 1. Mathematica is doing the right thing; [by definition] so the mistake 
> is yours.

The common variant is to claim that it's a feature and not a bug. That 
tends to avoid placing blame one way or the other.


> 2. If you computed something else, different from what you wrote, you 
> would get a different answer, but corresponding to what you expected.
> 
> 3. How can you expect Mathematica to read your mind?
> 
> Here's one more standard, (from me, anyway).   Mathematica has a design 
> problem.
> 
> 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.).

Many here would agree with this diagnosis. We have endeavored to repair 
it (see below).


> 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.
> 
> RJF
> 

This already exists, as of version 7. One can specify 
Method->"Procedural". But it will not help for infinite or symbolic limits.

In[8]:= Sum[t[i], {i, 1, Infinity}, Method->"Procedural"]
Out[8]= 0

Possibly such cases should return unevaluated, with an error message.

Daniel Lichtblau
Wolfram Research



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