For real numbers it only makes sense to consider countable index sets. If you take an uncountable set of positive real numbers and you want to define their "sum", it turns out that it is always infinite.. and the argument is that you can always find a countable subset whoose sum diverges (in the usual sense)..
There are many ways to see this, and one of them is:
Suppose you have an uncountable set of positive numbers X.
Consider the following subsets of R^+ (the set of positive real numbers): [1, \infty[; [1/2,1]; [1/3, 1]; [1/4, 1] ... [1/n, 1] ...
These subsets cover R^+, so at least one of them ([1/n, 1] for example) has an infinite number of elements of X. Now take a countable set of this infinite set and consider its series. All its elements are greater than 1/n, so the series diverges in the usual sense.
Yeah, an entry about this would be nice.. If I have time and nobody else does it I'll write it later today. |
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