The first time I ever heard about the number $e$ was through compound interest. That is, the money you owe will be multiplied by $e$ each year if it compounds infinitely. Mathematically, we write
$$e = \lim_{n\to\infty}\left(1 + \frac{1}{n}\right)^n.$$
In a calculus setting, however, you generally first see $e$ in the fact that the function $e^x$ is its own derivative, and that $ce^x$ is the only family of functions satisfying this property. When you find the Taylor series of $e^x$, this fact leads you to the following infinite summation for $e$:
$$e = \sum_{n = 0}^\infty \frac{1}{n!}$$
These two facts, independently, make a lot of sense to me. However, I cannot figure out how to equate the two expressions in my mind; they seem entirely different. One is a infinite sum, and one is a limit. One has factorials and one has exponents. I can't picture any mathematical manipulation, besides sheer coincidence, that would allow you to manipulate one expression into the other.
So my question is:
Why does $$\displaystyle\lim_{n\to\infty}\left(1 + \frac{1}{n}\right)^n = \displaystyle\sum_{n = 0}^\infty \frac{1}{n!}?$$
All I'm looking for is a proof that the two expressions are equivalent via mathematical means; while intuition would be nice as well, it's not necessary for an answer.
Thanks for your explanation!
