The issue has nothing to do with infinity or 1/infinity. It has to do with the definition of real numbers and what it means for a sequence (or series) to converge.
Here's a sketch of the ideas as I see them.
1. Define the real numbers; for the sake of argument, as Cauchy sequences of rationals.
2. Define when two real numbers are equal: Suppose A=(an) and B=(bn) (Cauchy sequences). Then A = B if the terms of the sequence (an-bn) can be made less than any epsilon>0 by making n big enough.
3. Define the arithmetic of the reals: addition, multiplication, inequalities, absolute value.
4. Define L = Lim An if |L - An| can be made less than any epsilon>0 by making n big enough (L and the An are reals here -- see 3 above).
5. Define the infinite sum: Sum(An) = L if L is the limit of the partial sums of the An (see 4 above).
That all there is! No mystery. The wonderful thing about the real analysis developed by Weierstrasse, Cauchy et. al. is that it FREED US FROM THE TERROR OF THE INFINITE! You never have to mention infinity anywhere in definitions 1-5 above. It is very worthwhile to read Aristotle on the distinction between the "actual" infinite (which is scary and not quite believable) and the "potential" infinite, which is what modern (post 18th century) analysis is all about. Forget "nonstandard analysis" which is based on formalism and, as they say, seeks to explain the obscure by the more obscure.
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