One dollar is invested with an annual interest rate r$.$ If the investor cannot withdraw the principal before the period ends once deciding to invest, what will he obtain after the first year $($and then after t years$)?$

One dollar is invested with an annual interest rate r$.$

$1.$ If the investor cannot withdraw the principal before the end ends once deciding to

invest, what will he obtain after the first year $($and then after t years$)?$

From now on$,$ assume that the investor can withdraw the principal plus the interest and reinvest at the same rate at any time.

$2.$ In case r $=1,$ calculate the amount obtained after one year if the investor withdraws and reinvests every six months, three months and one month.

$3.$ In case r $=1,$ write a formula for the amount obtained after one year if the investor withdraws and reinvests n times during the year. Using a spreadsheet, convince yourself that this amount converges to a finite number when the investor acts more and more frequently $($Hint: this finite number is the base for natural logarithm e $2\mathrm{.}718281828$