Compound Interest

Because $\left(1+\dfrac rn\right)^{nt}$ is exactly the compound-amount formula $A=P\left(1+\dfrac rn\right)^{nt}$ (principal $P$, rate $r/100$, $n$ compounding periods per year, $t$ years), it is natural to ask what happens as the number of compounding periods per year grows without bound (quarterly $\to$ monthly $\to$ daily $\to$ every minute $\to \cdots$).

Fixing $P=1,\ r=1,\ t=1$, define $A_n=\left(1+\dfrac1n\right)^n$ and tabulate:

As $n$ grows, $A_n$ approaches a fixed irrational number, $e\approx2.718281828\ldots$ (Euler's number). The compound-interest formula in the limit of continuous compounding therefore becomes

$A=Pe^{rt},$ …