As a topic for a lesson, it would be interesting to see how sums invariant of rearranging and absolutely convergent sums don't coincide in infinite-dimensional Banach spaces
Just a note: reordering only a finite number of terms preserves the limit. I'll provide a rigorous proof. Let sum(an)=c, let the partial sums of (an) be called (pn), and let (bn) be the sequence (an)with a finite number of reordered terms and partial sums called (qn). That is, there are a finite number of indices where (an) doesn't equal (bn). Thus there is a maximum index k-1 where (an) and (bn) differ, so an=bn for all n>=k. Since finite sums stay the same after reordering, and the set of the first k terms of both sequences are the same, the sum of the first k terms of (an) equals the sum of the first k terms of (bn). So, pk=qk and, by induction, qN=q(N-1)+bN=p(N-1)+a(N-1)=pN for all N>k too. It should be obvious from here, but I always enjoy showing the limit definition. Since sum(an) converges to c, we have that (pn) converges c. By the limit definition, this is equivalent to saying that for all r>0 there exists an index h such that |pn-c|=h. Of course, max(k,h) is greater than or equal to both h and k, so |pn-c|=max(k,h). That is, |qn-c|max(k,h). Since such an h exists for each r>0, (qn) converges to c by the limit definition and sum(bn)=c.
