After some research I have found out that in order to find the power series expansion of a function I can apply Taylor's formula. In order to prove Taylor or Maclaurin formulas I need to assume power series as valid. Eventually I cannot find a proof to understand why some functions such as $exp(x)$ or $sin x$ can be expanded using power series, one that will not assume Taylor as valid, while Taylor uses power series. My mind is struggling in a circular argument. I've also checked similar questions here and they use the same technique. Any help?
PS: I found out in Tom Apostol- Calculus, Vol I, that the proof is linked to a circle of absolute convergence, but it is not detailed
PS2: To clarify, I am looking for a proof for power series. Why can some functions be written as power series, and if possible, one that does not make use of Taylor's expansion.