**The compound-interest problem** Jacob Bernoulli discovered this constant by studying a question about

compound interest.

One simple example is an account that starts with $1.00 and pays 100% interest per year. If the interest is credited once, at the end of the year, the value is $2.00; but if the interest is computed and added twice in the year, the $1 is multiplied by 1.5 twice, yielding $1.00×1.5² = $2.25. Compounding quarterly yields $1.00×1.254 = $2.4414…, and compounding monthly yields $1.00×(1.0833…)12 = $2.613035….

Bernoulli noticed that this sequence approaches a limit (the

force of interest) for more and smaller compounding intervals. Compounding weekly yields $2.692597…, while compounding daily yields $2.714567…, just two cents more. Using

*n* as the number of compounding intervals, with interest of 1⁄

*n* in each interval, the limit for large

*n* is the number that came to be known as

*e*; with

*continuous* compounding, the account value will reach $2.7182818…. More generally, an account that starts at $1, and yields (1+

*R*) dollars at simple interest, will yield

*e**R* dollars with continuous compounding.