Are E(L) the expected bets? Then the first line (Lard wins the first round) should not be zero as he still has to put his bet.
Is E(L) the expected total result? Then the first line (Lard wins the first round) should not be zero as he gains money.

The other lines look wrong as well for the same reason.

It's not the expected bet, but the expected loss. Each line is the probability for a certain outcome multiplied by the amount of money that would be lost in said outcome.

So in the first line, Lars wins on the first try and loses nothing. In the second line, Lars loses on the first try but wins on the second, losing one dollar. and so on until Lars loses 5 times in a row and gives up.

However, I get the wrong expected loss of money with this approach.

Ah yes, of course. He doesn't get his starting money back even if he wins (he just gets twice the original amount). That's where I was stuck. OK, i get the correct answer now. thanks