Why do we list conditional convergence of a series as a special case? We are covering power series and when determining convergence for X. The teachers wants to isolate conditional convergence and his answers seems vague to me. What I’d like to know is that if conditional and absolute convergence both lead to convergence, why do we specify that at all?
 A: The important thing about conditionally convergent series is that if you change the order of the terms in a such a series, you can make the series converge to anything you want.
Think about it this way: rearranging terms of a series changes the partial sums. So it makes sense that doing so could change the limit of the partial sums.
A bit rigorously:
Definition. A series is called conditionally convergent if $\sum a_n$ converges but $\sum |a_n|$ diverges. Alternatively, a given series is conditionally convergent iff the series of positive terms diverges and the series of negative terms diverges.
Remark. It can be shown that if $\sum |a_n|$ converges, then every rearrangement of the series $\sum a_n$ has the same sum.
Riemann's Theorem. A conditionally convergent series (of real numbers) can be rearranged to sum to any real number.
Proof. Assume we have a conditionally convergent series. Since the given series converges, the terms of the series get small.
Since the given series is conditionally convergent, the series of positive terms
diverges and the series of negative terms diverges (by definition)
Given S, we want to form a rearrangement of the series that sums to $S$. To begin, select enough of the positive terms (choosing largest first) until the partial sum is larger than $S$. This is possible because the divergence of the positives gives us an infinite amount of terms to choose from.
For the next terms of the series, choose enough of the negative terms (choosing the most negative terms first) to make the partial sum less than $S$.
Continue, choosing the positive and negatives terms in order of decreasing
size so that the partial sums swing to more than S, then less, then more, etc. Since the terms of the series get small, the oscillations get smaller and smaller so that the sequence of partial sums of the rearranged series converges to $S$.
(Source: I originally found this proof here when I was learning about conditional convergence myself.)
