As said in https://math.stackexchange.com/a/730227/441785, is there any theorem regarding why a non-smooth (not indefinitely differentiable function) results in a infite sum?
Does this mean a smooth function be represented by a finite sum, as in a Fourier Series, for example?
Any help is greatly appreciated :)
As stated in the comments and the other answer, $A\cos at$ and $B\sin bt$ ($A, B, a, b$ constants) are both smooth. A sum of two smooth functions is also smooth. So any finite sum of functions of the form $A\cos at$ or $B\sin bt$ for various constants, will be smooth. Finite Fourier series are examples. So the Fourier series of a non-smooth function has to be infinite.
Quite obviously it does not mean any such thing. That is the converse of the statement, and there is no reason to expect the converse of a true statement to also be true:
"If it is raining, the ground is wet." is true (within reasonable interpretation). The converse is "If the ground is wet, it is raining." is easily seen to be false. Just ask the guy who runs his sprinklers all day.
It can be that the converse of a true statement is also true. The converse of "if $a = 1$, then $a+1 = 2$" is "if $a + 1 = 2$, then $a = 1$", and both are true. But that is not the case here.
For example $e^{\sin t}$ does not have a finite Fourier expansion.