Are there derivations of the taylor, fourier and laurant series?
Of course. They didn't just fall out of the sky. Specifically what do you mean?
In all three cases I know how if you accept that they can be written in that form (I.e. as power series or infinite series of the sines and cosines), then you can derive the coefficients using cleverly picked transformations, i.e. differentiation, the fourier transforms or Cauchy integral formula trick. What I don't know is how you derive the original bit.
That's where 'inspiration' comes in.
The ideas behind these series representations stem from the earliest days of mathematics. In the 17th century, it was established that an infinite series could converge to a finite result. Once this property of infinite series was established, then series could be applied to other areas of mathematics.
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