Please note that English is not my first language, so I apologize for any unprecise vocabulary.
Through my scolarity, I have seen 2 main definitions for the exponential function (apart from e^x): one uses a limit, the second a power series.
$$\exp x=\lim_{n\to+\infty} (1+\frac{x}{n})^n$$ $$\exp x:=\sum _{k=0}^{\infty }{\frac {x^{k}}{k!}}$$
I understand the intuition of the first formula (continuous compound interest), but I have no intuition or way of understanding for the second formula, which is, if I'm not mistaken, the most used. People, whenever they wanted to prove something related to the exponential function, used either of those definitions (the easiest one to execute the proof with), and moved on. For instance, e^x is equal to this power series because e^x is its own derivative, and when asked why it's its own derivative, we are answered that since it's equal to its power series, then it must be its own derivative.
I was never really explained why those 2 formulas worked and were essentially equivalent to e^x (for real numbers). So what I understood from those explanations I was given was basically "Let's prove 2+2=4, okay so since 2+2=4 then we can deduce 4=2+2, and therefore 2+2=4".
So, how can we, step by step, without using circular reasoning or "assumed facts", find the number e and the exponential function, as well as some of its main properties? Basically, where to start? Thanks in advance.
