Braindead Way to Derive Taylor Series of Exponential Function

$e^x$ was first defined by a formula for continuous compound interest:

Let’s use this to get that $e^x=\sum _{n=0}^N\frac{x^n}{n!}$

First, remove the limit and replace it with an unlimited¹ integer $N$.

That gives us the expression $(1+\frac{x}{N}{)}^N$. Which is just a polynomial with an unlimited number of terms. We can expand it by the binomial theorem.

$1^{N-n}$ is just 1, so we can drop it because it’s the identity for multiplication. Which gives us

We can expand $(\genfrac{}{}{0ex}{}{N}{n})$ by using its definition and move the $n$ exponent into each term.

Cancel $(N-n)!$ from $N!$ to get

Now for the only real bit of cleverness. N is unlimited, so $\frac{N-k}{N}$ where k is limited is infinitely close to 1. Infinite closeness is an equivalence relation, which is almost as good as true equality. In this case, we use it to cancel out $\frac{N}{N},\frac{N-1}{N}\dots \frac{N-n+1}{N}$. This leaves behind

I read Euler recently and it turns out this is how he found the Taylor Series in the first place².