## Continuously Compounded Interest

If an amount of money is invested so that compound interest is accrued at the rate r% per time period, then after time periods the amount of money will have grown to an amount If however, the interest is compounded more regularly, then something a little bit strange happens. Suppose £1000 is invested at 12% per annum. If interested is compounded annually then at the end of a year, the original £1000 will have grown to £1120. If however, it is compounded monthly, then the monthly rate of interest will be 12/12 =1% and after 1 year the original £1000 will have grown to In fact if the year is divided into time periods, so that interest is compounded n times a year, the interest per time period is and the amount of money will have grown to The table below shows the investment after 1 year for various values of n.  10 1126.691779 100 1127.415743 1000 1127.488731 10000 1127.495196 100000 1127.495975

As tends to infinity, this expression tends to a limit We can generalise this reasoning, so that if annual interest of is compounded continuously on an investment of at the end of a year the investment will have grown to and at the end of years the principal will have grown to 