The Money Never Runs Out

Fred is an incredibly talented investor: so talented, in fact, that he doubles the amount of money in his bank account each year.

Fred doesn't worry about running out of money, so he spends increasing amounts every year, and never makes any deposits other than his original deposit. At the beginning of the first year (right after the original deposit), he spends $1. At the beginning of the second year, he spends $2. At the beginning of the third year, he spends $3, and so on.

What is the minimum amount of money (in dollars) Fred needs in his original deposit so that he never runs out of money?


Note: As an explicit example, this is what would happen if the initial deposit was $20:

\[\begin{array}{|c|c|c|} \hline & \text{Account at the} & & \text{Account after} & \text{Account after} \\ \text{Year} & \text{beginning of year} & \text{Amount spent} & \text{spending} & \text{doubling} \\ \hline 1 & $20 & $1 & $19 & $38 \\ 2 & $38 & $2 & $36 & $72 \\ 3 & $72 & $3 & $69 & $138 \\ 4 & $138 & $4 & $134 & $268 \\ \hline \end{array}\]

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