Chapter 3.7, Problem 45E

If an initial amount A₀ of money is invested at an interest rate r compounded n times a year, the value of the investment after t years is

$A=A_0{\left(1+\frac{r}{n}\right)}^{nt}$

If we let n →∞, we refer to the continuous compounding of interest. Use 1’Hospital’s Rule to show that if interest is compounded continuously, then the amount after t years is

$A=A_0{e}^{rt}$