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A Gambler's Problem

Question

Let me present you with a game.  It's a gambling game.  The rules are pretty straightforward:

1. You begin the game with 10 US dollars

2. Every round, you may wager any of your current US dollars

3. After you have wagered, we toss a fair coin, which leads to one of two outcomes:
b.  Tails - You get your wager back AND win 200% of your wager's value

4. You have 100 rounds in which to maximise your profits as much as possible

5. It should go without saying, but if you ever end a round with 0 US dollars, you've lost

The game is obviously in your favour, but provide a betting strategy which uses these rules to win as many US dollars as possible.

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Spoiler

If you bet 10 USD in the first round, your expectancy is 20 USD. Consequently, in each round, you should bet all you money.

However, I would never bet on even 10 tails in a row.

Whatever your strategy is, there always is a chance you get bankrupt.

There is a similar problem: a coin is tossed and as long as heads come out, 2 USD are added to the bank. When tails are tossed, the game ends and you cash the bank. How much would you pay to play this game? (Remember that you might win an infinite amount of USD.)

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