St Petersburg with a finite bank

Idea. In the uncapped game every outcome contributes 1 dollar to the expectation, so the sum diverges. The cap cuts the sum after 20 terms and replaces the rest by a single term.

Step 1: the uncapped terms. The first head comes on toss $n$ with probability $2^{-n}$. For $n\le 20$ the payout is $2^n$, so each of these outcomes contributes $$2^n\cdot 2^{-n}=1,$$ for a total of $20$.

Step 2: the capped part. The first head comes after toss 20 with probability $2^{-20}$ (the first 20 tosses are all tails), and the payout is then $2^{20}$. This contributes $$2^{20}\cdot 2^{-20}=1.$$

Step 3. The expected payout is $20+1=21$ dollars.

The lesson. The uncapped game has an infinite expected value, yet nobody would pay much to play it. With any real bank, the value grows only like the logarithm of the bank: a bank of $2^{30}$ dollars, about a billion, raises the fair price only to 31 dollars.