Alice promises to pay Bob $1000 today, $2000 in one year, $3000 in two years, and so on indefinitely. Assume that the interest rate is 2%, compounded continuously. (a) Express the total present...

Alice promises to pay Bob $1000 today, $2000 in one year, $3000 in two years, and so on indefinitely. Assume that the interest rate is 2%, compounded continuously.

(a) Express the total present value of Alice’s promise to Bob as an indefinite integral, and evaluate it.

(b)  In approximately what year will Alice make the largest payment to Bob, in terms of present value?

Expert Answers
justaguide eNotes educator| Certified Educator

Alice promises to pay Bob $1000 today, $2000 in one year, $3000 in two years, and so on indefinitely. The interest rate to be used is 2%, compounded continuously.

The amount given by Alice to Bob at the beginning of year n is given by (n+1)*1000.

As interest is compounded continuously and the amount is received at the beginning of the year, the present value of an amount received after n years is A = (n+1)*1000/e^(0.02*n)

The total present value of Alice's promise is

`1000+ sum_(n=1)^(oo)(n+1)*1000/e^(0.02*n) `

= 2.54942*10^6

Alice would make the largest payment to Bob approximately at the start of year 49.