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Re: error with Sum and Infinity

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.

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 

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.


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