*An* interesting physical interpretation I've seen (i.e. of a particular infinite series) was the case of stacking rectangular blocks on top of each other.
Although difficult to explain in words, the question that motivates the resultant infinite series is "is it possible to stack identical blocks on top of each other such that, at some point, the n-th block does not overlap with the first (e.g. if the blocks have length 1, and the coordinate system is drawn such that the first block extends from the origin to x=1, is it possible to stack additional blocks on top of that first block such that eventually a block will be located entirely to the right of x=1, and is, so to speak, "not directly supported").
One can model this situation w/ some basic knowledge of centre of mass and the result is an infinite series. The question then becomes - does the series converge or diverge? If the series converges, then there is a finite horizontal extent to which stacked blocks can stretch out. If the series does not converge, then (in principle), one could start with a single block and create a stack where the top block is arbitrarily far away (in horizontal distance) from the bottom block.
In this case, it turns out the series diverges. However, the height of the stack grows much faster than its horizontal extent - both go to infinity, but the latter much more slowly than the former.
Not sure if this is the kind of thing you were thinking of, but I personally found it a very enlightening approach to limits (& I suspect there are many others ...).
Cheers,
Rax
