Any curve can be represented as a sum of sinusoidal curves, i.e. a Fourier series.
Not quite. Any
periodic curve can be represented as a sum of sinusoidal curves. The trick is in knowing the periodicity of $=f(t). protip: that f() ain't periodic.
Nope.
An infinite fourier series can match any continuous differentiable function.
Ex: an infinite series can add to a single square wave pulse.
A finite sum can approximate any such function as accurately as you want.