For one thing, a differential is something that can be integrated. Secondly, differentials provide essentially a generic way of writing down the chain rule. These two ways of looking at the concept of the differential seem very different from each other, and it seems at first astonishing that the same concept serves both purposes. The reason that this is so is simply that the method of integration by substitution is nothing but the chain rule written backwards. The point of view given above still applies, but integration has to be understood in the sense of line integrals, which I think students often find fairly intimidating and non-intuitive.
Most convergence tests are tests for positive series.
Even though many of the most useful series are not positive, these convergences tests still apply, since they can be used to test for absolute converence, which implies simple convergence. A positive series converges if and only if it is bounded. Restated, this says that either a positive series converges or its limit is infinity. It is easy to see why a series which converges absolutely must converge in the ordinary sense. It is also easy to see why the limit of an absolutely convergent series is not changed if one rearranges the terms.