Math Adventures/Compound Interest

In 1683 Jacob Bernoulli considered the following problem:

An account starts with $1.00 and pays 100 percent interest per year. If the interest is credited once, at the end of the year, the value of the account at year-end will be $2.00. What happens if the interest is computed and credited more frequently during the year?

Calculate the year-end value of the account if the interest in credited twice per year.

• Once per month?
• Once per week?
• Once per day?
• Every hour?
• Continuously?

Do you recognize that number?

If there are n compounding intervals, the interest for each interval will be 100%/n and the value at the end of the year will be $1.00 × (1 + 1/n)ⁿ

For continuous compounding the formula becomes:

$${\displaystyle \lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}}$$