Let me preface this by saying that I know I'm doing something wrong, I'm just here to find out what exactly.
We know that integrating a function yields constants. Something like $e^x$ yields $+c$ when integrated. It doesn't matter what arbitrary value of $c$ we choose, because when differentiated, it just goes away. When integrated multiple times, we get more constants, and powers of $x$ start popping out as well. So, if we integrate $e^x$ 3 times, we would get something like $e^x+c_1x^2+c_2x+c_3$, which, regardless of the constants, when differentiated 3 times would yield $e^x$ again. Said differently, we can say that the $(n+1)$-th integral of $e^x$ is equal to:
$$e^x + \sum_{0}^{n}a(n)*x^{n}$$
As $n$ goes to infinity (we integrate $e^x$ infinitely many times), we get an infinite power series. My issue with this is that power series can represent functions. So, if we say for example that our infinite power series has constants such that $a(n) = \frac{1}{n!}$, as we go out to infinity, this power series represents a function ($e^x$). Ok, so if we integrate $e^x$ infinitely many times, we get:
$$\lim_{n \to \infty} e^x+\sum_{0}^{n}\frac{x^n}{n!}$$
where the power series is equal to $e^x$. Making this substitution yields $e^x + e^x$, which is just $2e^x$. Ok, so infinitely integrating $e^x$ yields $2e^x$. Obviously wrong. Moreover, it doesn't matter how many times I differentiate $2e^x$, I can do it infinitely many times, and I'm still never getting back to $e^x$. So where is the mistake? Integration yields constants. I am allowed to choose the value of my constants (because they don't matter if I differentiate them away). However, if I choose constants such that as I integrate out to infinity, the constants times the powers of $x$ create the power series of a function, I feel it ought to be legal to simply substitute in that function. But then, no amount of differentiation will yield the original function. So, what gives? I feel like it has something to do with the fact that I can't just choose my own constants, but I'm arguing that it doesn't matter, because $n$ integrations should come undone by $n$ differentiations, regardless of my constants.