How to resolve the harmonic series paradox presented in this video by James Tanton?

If your concern is the apparent paradox about a infinite length of paint and a finite area, then you might want to consider what Wikipedia says on about Gabriel's horn with an infinite area of paint and a finite volume, and then take it down a dimension:

Since the Horn has finite volume but infinite surface area, it seems that it could be filled with a finite quantity of paint, and yet that paint would not be sufficient to coat its inner surface – an apparent paradox. In fact, in a theoretical mathematical sense, a finite amount of paint can coat an infinite area, provided the thickness of the coat becomes vanishingly small "quickly enough" to compensate for the ever-expanding area, which in this case is forced to happen to an inner-surface coat as the horn narrows. However, to coat the outer surface of the horn with a constant thickness of paint, no matter how thin, would require an infinite amount of paint. Of course, in reality, paint is not infinitely divisible, and at some point the horn would become too narrow for even one molecule to pass.

• You beat me to it! – Cheerful Parsnip Nov 4 '12 at 23:26

In general, the length of a line and the area of a curve over that line do not have to both be infinite, or both be finite. It is not a paradox. See improper integrals. http://en.wikipedia.org/wiki/Improper_integral. Note that length and area do not have the same dimensions.

The infinite amount of ink is due to the fact that you draw with a pen of constant width.

But when painting the squares, the width gets smaller and smaller, at some point smaller than the ink stroke, further down to zero. This allows a finite amount of paint.

There is no paradox here. The length of the ink needed to color the real line is equal to the length of the ink needed to color those squares and both lengths are infinite but area of the ink needed to color those squares is finite. The paradox arises because the author talks in such a way that he does not mention (or I did not hear it) that when coloring the real line the ink needed is measured as length and when coloring the squares the ink needed is measured as area.