Check your intuition

Suppose Charlotte and Beatrice play a coin-flipping game with biased coins. The game iterates as per this rule:

Start at step 1. At step n, flip a coin with \(\frac{1}{2^n+2}\) probability of coming up heads and \(\frac{2^n+1}{2^n+2}\) probability coming up of tails. If heads comes out, Beatrice wins. Else, proceed to step n+1.

What is the probability that Beatrice wins?

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