Simple examples are really encouraging in the understanding of rearrangements of infinite series, since many texts and teachers provide only existence theorems. In the absence of examples, an existence theorem is just a statement and lends little confidence to understanding. Iterated sums of double series seem to have a similar spirit of rearrangements of series though they are not the same. Therefore learning about iterated sums of double series will help the understanding of the rearrangements of series. Without simple examples, which is the case with many texts, theorems about iterated sums do not contribute much to our understanding. Many students and perhaps some teachers would be discouraged by this in the learning process. In this note, several inspiring examples are presented to show that infinite sums indeed behave very differently from finite sums.