exp_ln - The number e and natural logarithms An...

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The number e and natural logarithms An introduction to the mathematical constant e via compound interest If P dollar is invested with an interest rate of r % compounded annually, it grows to P + 1 r 100 n dollars after n years. For example, if $100 is invested at an interest rate of 10%, the amounts in dollars produced after 8 successive years to the nearest cent are given in the following table. Number of years | 0 1 2 3 4 5 6 7 8 Amount in dollars | 100 110 121 133.1 146.41 161.05 177.16 194.87 214.36 Now suppose that loan shark A offers a loan of L dollars for a period of one year at an annual rate of 100% interest, so that 2 L dollars must be repaid at the end of the loan period. A greedier loan shark B offers the same annual interest rate but compounds the loan on a monthly basis with an effective annual interest rate of 100%. This means that, at the end of each month, the amount owing is multiplied by + 1 1 12 . The amount which must be repaid to B at the end of the year is therefore + 1 1 12 12 times the amount of the loan. + 1 1 12 12 ~ 2.61304. An even greedier loan shark C offers the same annual interest rate as A and B, but compounds the interest weekly. The amount to repaid to C would be + 1 1 52 52 times the amount of the loan. + 1 1 52 52 ~ 2.69260. This is not a huge amount more than would have to be repaid to loan shark B. Is there a limit to what can be obtained by shortening the period over which the interest is compounded, while the effective annual rate is kept at 100% ? Let's calculate the amount to repaid if the interest is compounded daily when the effective annual rate is 100%.
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This note was uploaded on 11/12/2011 for the course MATH 111 taught by Professor Uri during the Spring '08 term at Rutgers.

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exp_ln - The number e and natural logarithms An...

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