Price a coin-doubling game rationally

Consider this game: start with $1. Flip a fair coin repeatedly. On each Head, your current bankroll doubles; on the first Tail, the game ends and you take your current bankroll. Example: H,H,T → you take $4. a) Compute the expected payout and state whether it converges; show your derivation. b) If you must state a maximum price you would pay to play once given risk‑averse logarithmic utility U(w)=ln(w) and initial wealth W, what price do you choose? c) If you can play repeatedly with Kelly staking fraction f of wealth against a fixed ticket price p per play, derive the condition on p and f for positive expected log‑growth and give the optimal f*. d) Discuss how a house cap (maximum payout C) changes the expected value and your reservation price.