St. Petersburg lottery[1] has an infinite expected value, but it definitely is not an investment. No actual person would be willing to "invest" more than a very small amount into it.
1: there is a finite upper bound to the payoff, fixed at the amount the casino has, so calling the ev infinite ignores real constraints.
2: infinity is a valid concept on Mathematics, but not in economics. If there were infinity in economics there would be no constraints and no reason for the field of economics to exist.
3: by my definition, playing the st petersburg lottery is an investment up to the amount of of money the casino has on hand. I'd argue that it is further prudent to play if you're guaranteed a sufficient N and $x price such that your available disposable cash = ($x * N) leads to a positive value at least 50% of the time.
