Here is my attempt to understand this:
Let $$ \sum_{n=1}^{\infty}\left(-1\right)^{\left(n+1\right)}a_n $$ be an alternating series
now the infinite sum is defined as the limit of the sequence of partial sums and so a partial sum of $m$ terms of the series where $m$ is an even number (I think it wouldn't make a difference if $m$ was odd since we will take the limit as $m$ approaches infinity anyway) is:
$$ \sum_{n=1}^{m}\left(-1\right)^{\left(n+1\right)}a_n = \sum_{n=1}^{m/2}(a_{2n-1}) – \sum_{n=1}^{m/2}(a_{2n})$$
Now taking the limit as $m$ approaches infinity on both sides and denoting the limit on the left by $S$ , the first sum on the right side by $S_o$ and the last sum by $S_e$
$$S = \lim_{m\rightarrow\infty}(S_o-S_e)$$
the right term is the limit of a difference , If limits as $m$ approaches infinity of both $S_o$ and $S_e$ exist the limit can be separated which means that no matter how you change the order of summation of the original sum it can be rearranged such that the positive terms are together and the negative terms are together and since both limits exist their values are unique and their difference is the unique sum of the whole series.
this seems to make sense until I try to convince myself that if one (or both) of the limits of $S_o$ and $S_e$ as $m$ approaches infinity doesn't exist (since proving that one doesn't exist implies the same of the other) corresponds to proving the fact that changing the order of summation of the original series changes the value of $S$ and that it isn't unique anymore. I stop here with no idea how to proceed.
Thanks in advance
Best Answer
A sum $A$ of an absolutely convergent series does not depend on the order of summation because a main contribution to $A$ is made by (finite) heads of the series, for which we can apply the commutativity of addition, whereas sums of tails of the sequence can be arbitrarytiry small, see a proof.
Conversely, if a series converges, but not absolutely then a sum of its chosen members can tend to infinity, but the fixed order of the series prevents infinite partial sum growth by its collapsing, for instance, in a series $\sum_{n=1}^{\infty}\left(-1\right)^{\left(n+1\right)}\tfrac 1n$. If we relax this order then we can force the rearranged series to converges to an arbitrary real number or to diverge, see Riemann series theorem.