Deriving equations from fourier series representations

Say you have the coefficients [itex]a_k[/itex] of a Fourier series representation of some function [itex]x(t)[/itex]. You can easily then give [itex]x(t)[/itex] as

[tex]x(t) = \sum_{k = -\infty}^{\infty} a_k e^{i k \omega_0 t}[/tex]

But this doesn't do much good in telling you what the actual function looks like. For example, if we have

[tex] a_k = \frac{ \sin \left(k \pi / 2)} {k \pi} [/tex]

we can write [itex] x(t) [/itex] as

[tex]x(t) = \sum_{k \neq 0} \frac{ \sin \left(k \pi / 2)} {k \pi} e^{i k \omega_0 t}[/tex]

but you would have a hard time telling that this is a square wave with a duty cycle of 50% unless you've previously derived the series coefficients for that exact function.

Basically, my question is: is there a way to derive a more intuitive representation of a function given its Fourier series representation?
 

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