Geometric mean of many uniforms

Let $X_1,X_2,\dots$ be i.i.d. $\text{Unif}(0,1)$ and set $Y_n=\left(\prod_{k=1}^n X_k\right)^{1/n}$. The sequence $Y_n$ converges almost surely to a constant of the form $e^a$. What is $a$?

$-1$. Taking logs, $\ln Y_n$ is the average of $\ln X_k$. Since $-\ln X_k\sim\text{Exp}(1)$, $\mathbb{E}[\ln X_k]=-1$, and the strong law of large numbers gives $\ln Y_n\to-1$, so $Y_n\to e^{-1}$.