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*countably*infinite number of terms, yet it is supposed to be giving the under area under a curve which varies in a

*continuous*manner. Has anyone else thought seriously about this?

I suspect it is closely related to the fact some functions on a finite interval can be represented by a fourier series.

[tex]f(x) =\frac{a_0}{2}+\sum_{n=1}^\infty\left[a_n cos(nx)+ b_n sin(nx)\right][/tex]

Here we see that a function f consisting of a continuum of points [tex]f(x)[/tex] is represented entirely by a countably infinite set of numbers [tex]\{a_0,a_1,...,b_1,b_2,...\}[/tex].

I'm not able to pose a coherent question about this. There is something interesting here but I can't get my head around putting it into words.

I'm interested in thoughts anyone cares to share on continuum vs countable. Is there a relation between integrable and able to be represented by a Fourier series?