Roll a 20-sided die...

Every time I see a d20, my thoughts go back to some awesome dungeon encounters in a friend's basement in Lexington, Kentucky decades ago.  But today, the die is used for a math puzzle:

We play a game. You roll the die, and can elect to bank [that number of $], or roll again. If you bank, you walk away with the dollar amount shown on the die, and the game ends. If you elect to re-roll, it costs you $1 for each new roll. You can re-roll as often as you like. (Your first roll is free)...

What is the optimal strategy to maximize your winnings? If you follow this strategy, what is your expected return?...

It's pretty clear that if we rolled a natural 20 at any time, we'd instantly stop; We're never going to get better. It's not a stretch to see that, if we rolled a 19, we'd also stop; The only number that could beat a 19 is a 20, and we'd have to pay an additional $1 just for the privelage of rolling to see about getting it. Even if we did roll a 20 (a small chance), we'd net out with no gain, so why bother? For 18, it's less clear; we have a one chance of breaking evening, one chance of improving, and lots of chances of losing more, if we rolled again.

Hint:  start with this, and simplify -

Or... see the answer at the link (it's not intuitively obvious).