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I keep reading in multiple sources that amplifier output can be given by

V

_{out }= a

_{0}+ a

_{1}v(t) + a

_{2}v

^{2}(t) + a

_{3}v

^{3}(t) + .... + a

_{n}v

^{n}(t)

I've checked in three of my textbooks and there is not a clear definition (its often just stated) why this equation is used and why it works. I am not looking for an explanation of what each term is, but why it needs to be a power series and not something else?

I've searched on google and my textbooks, i'm looking for some mathematical vigour of how harmonics form

Please can someone advise who knows or point me in the direction of a good explanation?