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Unformatted text preview: the number
of compoundings per year to inﬁnity results in what is called continuously compounded interest.
Theorem 6.8. If you invest $1 at 100% interest compounded continuously, then you will have $e
at the end of one year.
In fact, the rate of increase of the amount in the account is exponential as well. This is the quality that really
deﬁnes exponential functions and we refer the reader to a course in Calculus.
Once you’ve had a semester of Calculus, you’ll be able to fully appreciate this very lame pun.
Or deﬁne, depending on your point of view. 6.5 Applications of Exponential and Logarithmic Functions 381 Using this deﬁnition of e and a little Calculus, we can take Equation 6.2 and produce a formula for
continuously compounded interest.
Equation 6.3. Continuously Compounded Interest: If an initial principal P is invested at
an annual rate r and the interest is compounded continuously, the amount A in the account after
t years is
A(t) = P ert
If we take the scenario of Example 6.5.1 and compare monthly compounding to continuous compound...
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