**1. Correct. The answer is false**

You should calculate the expected return using the definition of discrete expectation. In this particular case:

_{∞}

E(Z) = ∑ z(1/2)^{z}

^{z=1}

E(Z) = 1/2 + 2(1/4) + 3 (1/8) + . . .

This is an infinite sum of values less than 1. Hence, the sum converges to a finite value. To find this sum, let

S = 1/2 +2(1/4) + 3(1/8) + 4(1/16) + . . .

Then,

(1/2)S = (1/4) +2(1/8) +3(1/16) + . .

Subtracting (1/2)S from S,

(1/2)Q = 1/2+ (1/4) + (1/8) + (1/16)+ . . =1.

Thus

E(Z)= S = 2 ($ ten thousands).