>>11655938You're wrong with
>>11655931 because x1 and x4 are not defined. You've re-enumerated the quantities but not specified you've done so. One instance of a valid expansion (what you probably meant) is, of course, .
Note that since the indexing set for the sum is NOT necessarily being treated as ordered (the generic set curly braces are being used), there is an inherent ambiguity of the order in which the sum is to be carried out. Since the set is finite, all permutations of the sums will give the same quantity by the commutativity of the sum. For an infinite sum, you CANNOT, in general, rearrange the terms in the sum and expect to get the same answer.
In fact, Riemann proved that if a sum converges, but not absolutely, then given any -\infty <= a <= b <= +\infty, we can always rearrange the sum so that both the supremum of the partial sums approaches b and the infimum of the partial sums approaches a at the same time (i.e., the rearrangement achieves both things simultaneously). Further, Riemann showed that if a sum converges absolutely, then every rearrangement of the sum converges to the same value.
For uncountable sets, the sum can be defined to be convergent if the sum over all countable subsets (with ANY possible order on each countable subset in question) converges to the same value. By Riemann, this is equivalent to saying that the sum is absolutely convergent (to the same value) over every countable subset. Another way to define the sum is to take the limit of the net from the directed set of all finite subsets of the your indexing set (where the order is set inclusion) to the topological space in question (the reals, for instance).
