Back-to-back heads with coins of bias 1 over k

Kevin tosses $n\ge2$ coins in order, where coin $k$ lands heads with probability $\tfrac1k$. He earns $\text{\char36}1$ for every pair of adjacent coins that both show heads, with overlaps counting separately (so HHH earns $\text{\char36}2$). If $p(n)$ is his expected earnings with $n$ coins, find $\lim_{n\to\infty}p(n)$ (enter $-1$ if it does not exist).

$1$. By linearity, the expected payout is the sum over adjacent coin pairs $(k,k+1)$ of the chance both show heads, $\tfrac{1}{k}\cdot\tfrac{1}{k+1}=\tfrac1k-\tfrac1{k+1}$. The sum telescopes to $p(n)=1-\tfrac1n$, which tends to $1$.