In economy, it’s well known that compound interest at a constant interest rate provides exponential growth of the capital.
Why exponential though?
The general expression for the compound interest is just a geometric progression
If we call $r\in(0,1)$ the interest rate which applies after a time $t$, the capital as a discrete time function has the expression
$$C(nt)= C(0)(1+r)^n=C(0)\sum_{k=0}^n \binom{n}{k}r^k$$
How is the exponential obtained in the continuum limit?
Best Answer
The sum accumulated after a period $t$ using an annual interest rate $r$ and a compounding frequency of $n$ (i.e. quarterly, monthly) on an initial capital $C_0$ is:
$$ C_t(n) = C_0 \bigg(1+\frac{r}{n}\bigg)^{nt} $$
If the compunding frequency $n$ tends to infinity (we then imagine to compound infinitely often, i.e. continously) we have the exponential hrowth you mentioned. Indeed
$$ \lim_{n \to \infty}C_t(n) = \lim_{n \to \infty}C_0 \bigg(1+\frac{rt}{nt}\bigg)^{nt} = C_0 e^{rt} $$