If a principal of P is invested for t years at an annual rate of r and interest is compounded m times a year, then the interest rate per conversion period is rim and there are mt conversion periods. Thus if pound amount S at the end of t years becomes

S = P (I + r/m)^mt

Now we ask what happens to the amount as the number of times m that the interest is compounded per year gets larger and larger. When interest is so compounded, we say that interest is compounded continuously. Thus in the case of continous compounding, the amount S is given by


If we let r/m = x, then as m→∞, we have x→0.Thus



Hence the compound amount S of a principal P after t years at an annual interest rate of r compounded continuously is given by

S = Pe^rt

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