Example of a false proof when a Fourier series is not unique?

Question

I am attempting to come up with an example to illustrate why one should care that a function has a unique Fourier series expansion. Inspired by the fact that one can rearrange terms in a conditionally convergent series to obtain any number they want, I'm hoping that if one has a function with a non-unique Fourier series expansion, then one can do something with that series to obtain some kind of false result. Is anyone aware of such an example? Thank you.