TITLE: Definition of e
QUESTION [13 upvotes]: I'm very eager to know and understand the definition of $e$. Textbooks define $e$ as follows
$$ e = \lim_{p\to\infty} \left[1+\frac1{p}\right]^p \approx 2.71828 $$
Is there an "easy to understand" proof of this? I'm really looking for a derivation of this which is very intuitive and easy to comprehend.
By the way I'm watching this video lecture.

REPLY [2 votes]: It's really easy to show that $\dfrac{d}{dx} 2^x = (2^x\cdot\text{some constant})$, and that $\dfrac{d}{dx} 4^x = (4^x\cdot\text{some other constant})$.
At http://wnk.hamline.edu/~mjhardy/1170/homework/6th.pdf [later edit: Here's another (actually older) version, the link to which is still working: http://www.math.umn.edu/~hardy/1271/handouts/September.28.pdf] you will see in problem #2 that it's really easy to see that with 2, the "constant" is less than 1, and with 4, the constant is more than 1.
By this point, you'll be suspecting that for some number somewhere between 2 and 4, the "constant" will be 1.
That number is $e$.
That's one way to begin understanding what $e$ is about.
Now recall that the derivative of $f$ is approximated by $\dfrac{f(x+h)-f(x)}{h}$ when $h$ is small.  Think about what happens when $h = 1/n$, and that may shed some light on $\left(1 + \dfrac{1}{n}\right)^n$.