As we’ve said before, the real numbers are a topological field. The fact that it’s a field means, among other things, that it comes equipped with an associative notion of addition. That is, for any finite sum we can change the order in which we perform the additions (though not the order of the terms themselves — that’s commutativity).
The topology of the real numbers means we can set up sums of longer and longer sequences of terms and talk sensibly about whether these sums — these series — converge or not. Unfortunately, this topological concept ends up breaking the algebraic structure in some cases. We no longer have the same freedom to change the order of summations.
When we write down a series, we’re implicitly including parentheses all the way to the left. Consider the partial sums:
But what if we wanted to add up the terms in a different order? Say we want to write
Well this is still a left-parenthesized expression, it’s just that the terms are not the ones we looked at before. If we write , , and then we have
So this is actually a partial sum of a different (though related) series whose terms are finite sums of terms from the first series.
More specifically, let’s choose a sequence of stopping points: an increasing sequence of natural numbers . In the example above we have , , and . Now we can define a new sequence
Then the sequence of partial sums of this series is a subsequence of the . Specifically
We say that the sequence is obtained from the sequence by “adding parentheses” (most clearly notable in the above expression for ). Alternately, we say that is obtained from by “removing parentheses”.
If the sequence converges, so must the subsequence , and moreover to the same limit. That is, if the series converges to , then any series obtained by adding parentheses also converges to .
However, convergence of a subsequence doesn’t imply convergence of the sequence. For example, consider and use . Then jumps back and forth between zero and one, but is identically zero. So just because a series converges, another one obtained by removing parentheses may not converge.