EXAMPLE 7 Continuous Compounding

For $1000 at an annual rate of 10%, compounded times in the course of a single year, what is the balance at the end of the year?

This quantity gets closer and closer to as the number of compoundings increases. No matter how frequently interest is compounded—daily, hourly, every second, infinitely often (“continuously”)—the original $1000 at the end of one year cannot grow beyond $1105.17. The values after 5 and 10 years are shown in the lower rows of Table 21.2 (page 873).