euler's number

What is $e$?

e.g. I borrow money at an annual interest rate $r$, at a yearly compound rate: $$
\underbrace{ x(1) }{ \text{loan after 1 year} } = (1+r) \cdot \underbrace{ x(0) }{ \text{initial loan} }

but like this, banks would loose money, so they compound more often monthly compound rate: $\left( 1+\frac{r}{12} \right)^{12}\cdot x(0)$ this increases the total money by a teeny bit. but you can go even further daily compound rate: $\left( 1+\frac{r}{365} \right)^{365} \cdot x(0)$ in fact, they compound every instant / *continuously*:

x(1) = \underbrace{ \lim_{ n \to \infty } \left( \left( 1 + \frac{r}{n} \right)^{n} \right) }_{ \LARGE e^{r} } \cdot x(0)

The loan amount $x$ is continuously increasing at a rate $r$ proportional to the current loan amount… as a [[differential equations|differential equation]]:

\frac{dx}{dt} = rx \to x(t)=e^{rt}x(0)

> [!definition] Power series definition of $e^{at}$ > > $$ > e^{at} = \sum_{n=0}^{\infty} \frac{(at)^{n}}{n!} > $$ > [!sr]- What is $\lim_{ n \to \infty }\left( \frac{n-1}{n} \right)^{n}$ ? > >$$ > \lim_{ n \to \infty } \left( \frac{n-1}{n} \right)^{n} = \lim_{ n \to \infty } \left( \frac{n}{n} - \frac{1}{n} \right)^{n} = \lim_{ n \to \infty } \left( 1 - \frac{1}{n} \right)^{n} = e^{-1} = \frac{1}{e} >$$