A machine of a casino seems unconventional. The machine can be used only once and then replaced. The machine asks for a deposit of $2 and for the number of rounds the game is going to be played. Each round the game is played the machine saves the previously saved amount plus the square of the inverse of the number of times the game is going to be played. Finished the final round, the machine multiplies all of the saved amounts and pays this quantity to the player. Being allowed the infinity mode (which is a mode that plays infinitely many times), knowing that the player uses just one machine and that the machine saves $1 after the initial round - how much is going the player to collect by playing the infinity mode?