Doubling down on red

At an American roulette wheel (38 pockets, 18 of them red), you bet on red, which pays even money: win and you gain your stake, lose and it's gone. You bet $1. Each time you lose, you double your next bet: $2, then $4, and so on. You stop as soon as you win, or after 6 losses in a row, whichever comes first. Spins are independent. What is your expected profit, in dollars, for one such session?