If P dollars are invested at a yearly interest rate r per year, compounded m times per year for t years, then the compound amount A is

When m →∞, then the interest is compounded continuously. Use the following steps to derive a formula for the compound amount when interest is compounded continuously.

(a) Take the natural logarithm of both sides of the compound amount formula, and then show that

(b) Find the limit when m →∞. Verify that l’Hospital’s rule is applicable, and then show that the resulting limit is equal to ln A = (ln P) + rt.

(c) Solving the equation in part (b) for A, show that the compound amount when interest is compounded continuously at a rate of r for t years is A = Pe^rt.

Transcribed Image Text:

tm A 1 = P( 1 + ²)²™. m