medium Probability
A release-checked medium problem for training Counting With Indicators, Linearity of Expectation, Coins, Expected Value/LOTUS.
Brian flips a very strange coin $n$ times. This coin is crafted such that it has probability $1/k$ of showing heads on the $k$th flip. Brian receives $\$k$ if the $k$th and $(k+1)$st flips are both heads. Sequences are allowed to overlap, so the sequence $HHTHHHH$ yields a payout of $\$16$. As $n \rightarrow \infty$, what does Brian's expected payout converge to? If it converges to $\infty$, enter $-1$.
This Probability problem is tagged Counting With Indicators, Linearity of Expectation, Coins, Expected Value/LOTUS. State the random variables and conditioning information explicitly, then check the result against boundary cases before opening hints or a solution.
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