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Convergence
Posted:
May 9, 2001 10:33 AM
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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.
--- Mark
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