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).