So I saw some topics on this, but they didn't seem to answer exactly what I was looking for. Self-learning through Understanding Analysis, a exercise is the following:
So my misunderstanding is this: Abott proves this using the fact that putting parentheses around every n terms and treating them as a series of partial sums to add is equivalant to a subsequence:
However when I first tried the problem I didn't think of this rather obvious solution, and used the ( I believe) erroneous method of the associative property of addition. I've seen some examples of why this doesn't work, including the famoues +1-1+1-1... series. However I'd argue it's not the same because that series doesn't converge.
This is a very specific question I have so I'll try one more time to explain what I'm asking. Can't you prove the problem I posted by arguing that adding 1+.1+.01+.001... is the same as adding .1+.001+.01+1..., rather than what Abott does which is saying that it's the same as (1+.1)+(.01+.001)...
What I'm saying one more time:
$a_1+a_2+a_3+a_4...= Limit - \delta$
is the same as...
$a_3+a_2+a_4+a_1...= Limit - \delta$
(again, note this is not a sub-sequence because of order)
Thanks for any/all help!
I believe that you may be confused between associativity,ie. (a + b) + c = a + (b + c), and commutativity, which is a + b = b + a. When you reorder the series as you did in part 2, you actually need to use both of these properties.
What is proved in the link that you provided is associativity for infinite series that converge (Ie, you can group an infinite series in any manner and it is the same sum).
Your question asks if you can prove it by reordering and regrouping. What you are proposing is false for infinite series. There exist convergent series that converge to different values depending on the ordering. For examples, see http://en.wikipedia.org/wiki/Riemann_series_theorem. So your method is not correct, but the reason that it isn't correct is very interesting!