**Rearranging finitely many terms of an infinite series does not change the sum of the series** because there will be some partial sum \(s_n\) that contains all of the rearranged terms, \(s_n\) is a finite sum unaffected by rearrangement, all the partial sums \(s_k\) for \(k > n\) are thus unaffected by the rearrangement, and thus \(\displaystyle\sum a_n = \lim_{n\to\infty} s_n\) is unchanged by rearranging finitely many terms.

**Rearranging infinitely many terms of a conditionally convergent series can change the sum of the series!** When you rearrange infinitely many terms, you can change the value of infinitely many partial sums (and hence the sum of the series).

**Rearranging infinitely many terms of an absolutely convergent series does not change the sum of the series!**