The Swiss mathematician Daniel Bernoulli (1700-1782) was part of the famous Bernoulli family . In 1738 he published the book St. Petersburg Academy Proceedings in which the problem known as the St. Petersburg Paradox was first published. To understand this paradox, you must first work Problems 57-60 together. The paradox is seen after you answer the question in Problem 60.

Suppose you toss a coin and will win \(\$ 1\) if it comes up heads. If it comes up tails, you toss again. This time you will receive \(\$ 2\) if it comes up heads. If it comes up tails, toss again. This time you will receive \(\$ 4\) if it is heads. Continue in this fashion for a total of 1,000 flips of the coin, after which you receive nothing if it comes up tails. What is the mathematical expectation for this game?