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Application: Compound Interests

Example 7: Evaluating Limits

The limit limn→∞ (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 limn→∞ (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 aeb/l00 dollars after one year and to aebt/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 ∞,

limn→∞ an = ∞, limn→∞ bn = ∞, limn→∞ an/bn = ∞,

the sequence <an> is said to grow faster than the sequence <bn>. For each infinite H, both aH and bH are infinite. But aH/bH is still infinite, so aH is infinite even compared to bH.

Last Update: 2006-11-07