Let $\sum a_{n}$ be a complex series that converges. Now let $\sum a'_{n}$ be a rearrangement of that series. If we have $ \sum a_{n}=\sum a'_{n} $ for all rearrangements, is it true that $\sum a_{n}$ converges absolutely?
On a similar note, for all of you who own Rudin's Principles of Mathematical Analysis, can you check if there is a Theorem 3.56 in your book? Rudin cites it in Real and Complex Analysis, and I can't tell if it's a typo or was the theorem added in later printings.