Application: Compound Interests

Example 7: Evaluating Limits

The limit lim_n→∞ (1 + 1/n)ⁿ = e is closely related to compound interest. Suppose a bank pays interest on one dollar at the rate of 100% per year. If the interest is compounded n times per year the dollar will grow to (1 + 1/n) after 1/n years, to (1 + 1/n)^k after k/n years, and thus to (1 + 1/n)ⁿ after one year. Since lim_n→∞ (1 + 1/n)ⁿ = e, one dollar will grow to e dollars if the interest is compounded continuously for one year, and to e¹ dollars after t years.

More generally, suppose the account initially has a dollars and the bank pays interest at the rate of b% per year. If the interest is compounded n times per year, the account will grow as follows:

If the interest is compounded continuously the account will grow in one year to

We can evaluate this limit by setting x = (100/b)n, n = (b/100)x.

Thus the account grows to ae^b/l00 dollars after one year and to ae^bt/100 dollars after f years.

Sometimes we may wish to know how rapidly a sequence grows. If two sequences approach ∞ and their quotient also approaches ∞,

lim_n→∞ a_n = ∞, lim_n→∞ b_n = ∞, lim_n→∞ a_n/b_n = ∞,

the sequence <a_n> is said to grow faster than the sequence <b_n>. For each infinite H, both a_H and b_H are infinite. But a_H/b_H is still infinite, so a_H is infinite even compared to b_H.