1.4, due on September 12

This section felt a lot like a review of multivariable calculus, in that double summations are analogous to double integrals. I felt that the chapter made a lot of sense conceptually, however the examples still left me wanting. Some of the notation really made me think, as it didn't make much intuitive sense at first, but puzzling through the examples helped.
Something that really helped was thinking of it all in terms of infinite calculus; integrals are something I'm pretty familiar with at this point, and summations by themselves are also quite familiar, so partitioning the two and tackling them individually will probably go a long way.
One thing I was particularly curious about was whether there are any caveats with these finite sums in which the integral analogy begins to break down. That is, when do the familiar rules of integral calculus cease to apply? I know that infinite double sums can't be indiscriminately reordered (that was covered quite clearly in 341), but what else causes problems?