I think I am starting to understand.
So, a limiting process defines the ordering of an infinite sum which may be conditionally convergent unambiguously because the order is defined.
Taking your example limiting process of placing charges out to L, but then deciding or order the (finite) sum...
I think this might answer the question. If I understand you correctly, the situation proposed by Boas doesn't really model anything at all. Its aphysical to start with infinite discrete charges at infinite length. So, we get this conditionally convergent sum and have questions about how it was...
I understand, as you are pointing out, that with conditionally convergent series, you cannot reorder terms and this difficulty is caused by the infinite nature of the sum. The ordering is essential to the mathematics. What I do not understand is how you map a charge distribution to a ordering in...
Thanks again for taking the time to think this through with me.
What order do you apriori decide to add charges in? If there are charges at all the locations mentioned above, and each contributes a Force to the charge on the origin, how is it that you decide which order to compute the sum in...
Thank you for your answer.
The ordering of the terms effects the ultimate sum and convergence behavior. Had I choose a different order to compute in say, net postive + net negative, or two positive charges then a negative, no convergent answer would appear.
In cases of conditional convergence...
Mary Boas attempts to explain this by pointing out that the situation cannot arise because charges will have to be placed individually, and in an order, and that order would represent the order we sum in. That at any point the unplaced infinite charges would form an infinite divergent series...