Suppose we have a sequence of iid unif$(0,1)$ random variables. I want to know whether or not the sequence of $\mathbb E[\frac1{\bar X_n}]$ converge to 2 (since $\bar X_n$ strongly converges to $\frac12$). My guess is that it does, but I could not prove it.
Note that I have already proved that the expectation is infinite for $n=1$ and $4\log2$ for $n=2$ (using the fact that sum of iid uniforms is triangular), but couldn't proceed.