[2)(35 points) The next most basic geometric series formula is Tn(:t)=m+232+---+(n1)mquot;'1 =$S;(m) (2). Again, for the purpose of having specic...

Find a closed form expression for T5 by differentiating the formula in equa- tion (1) above.

(b)(5 points) Note that your formula in (a) is undefined when x = 1. You can try to use the usual form of L'Hopital's Rule to calculate the limit of T5(x) as x → 1, but it doesn't work this time. Why not?

(c)(10 points) You CAN use a suitable extended form of L'Hopital's Rule to calculate the limit. What do you get?

(d)(5 points) Calculate the limit as x → 1 of x + 2x2 + 3x3 + 4x4. Is it the same as when you got in (c)?

(e)(5 points) The ratio D5(x) = T5(x)/S5(x) represents the Macaulay Duration of the regular sequence of payments. The limit x → 1 corresponds to the limit as interest rate i → 0, in which all the payments have the same present value. Explain why D5(x → 1) makes sense as an average payment time.