A person plays an infinite sequence of games. He wins the th game with probability {\displaystyle 1/{\sqrt {n}}}, independently of the other games.

(i) Prove that for any , the probability is one that the player will accumulate [FONT=inherit dollars if he gets a dollar each time he wins two games in a row.[/FONT]

(ii) Does the claim in part (i) hold true if the player gets a dollar only if he wins three games in a row? Prove or disprove it.

I am posting again this problem.