Waiting time for a win with shrinking odds

In game number $n=1,2,3,\dots$ a winning number is drawn uniformly from $\{1,2,\dots,n+2\}$, and the player wins that game with probability $\tfrac{1}{n+2}$ (one of the $n+2$ values is his), independently across games. Let $W$ be the number of games played up to and including the first win. Find $E[W]$, or answer $-1$ if it is infinite.

$-1$ (the expectation is infinite). The chance of losing the first $k-1$ games telescopes: $\tfrac23\cdot\tfrac34\cdots\tfrac{k}{k+1}=\tfrac{2}{k+1}$. Then $E[W]=\sum_{k\ge1}P(W\ge k)=\sum_k\tfrac{2}{k+1}$ diverges like the harmonic series, even though $W$ is finite with probability $1$.