I know that for a (complex) series : If the series converge absolutely, then every rearrangement also converges and converge to the same limit.
But how to prove the converse : If every rearrangement of a (complex) series converge to the same limit, then the series itself also converge absolutely. That is to prove :
Let $\sum z_i$ be a complex series. If every rearrangement of $\sum z_i$ converge to the same limit, then $\sum z_i$ converges absolutely.