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