infinite envelopes

A game works as follows. There are infinitely many envelopes labeled 1, 2, 3, 4, ...

Inside envelope n is exactly n dollars.

You open envelopes one by one, starting from envelope 1.

After opening envelope n, you must immediately decide either:

1. Stop and take all the money you have collected so far, or

 

2. Continue to envelope n+1.

However, there is a catch.

Before the game starts, a random stopping envelope T is secretly chosen.

At the moment you open envelope T, the game ends automatically, and you receive nothing, regardless of how much you collected.

The stopping time satisfies:

P(T = n) = 2^-n for n = 1, 2, 3, ...

You know this distribution, but you do not know the realized value of T.

Question:

What stopping rule maximizes your expected total payoff, and what is that maximum expected value?