Alice and Bob each have a fair n-sided die. Alice rolls her die once. Bob then repeatedly throws his die until he rolls a number at least as big as the number Alice rolled. Each time Bob rolls, he pays Alice $1. (For example, if Alice rolls a 5, and Bob rolls a 4, then a 3, then
a 1, then a 5, the game ends and Alice gets $4. If Alice rolls a 1, then no matter what Bob rolls, the game will end immediately, and Alice will get $1.) Exactly how much money does Alice expect to win at this game? Prove that your answer is correct.