I'm considering such a question:
Suppose that there are $n$ i.i.d. variables denoted as $X_1,\cdots,X_n$, and a sequence is defined as: $a_n=\mathbb{E}\left (\frac{n}{\sum_{i=1}^NX_i}\right)$. It's easy to know that $a_n=\mathbb{E}\left(\frac{n}{\sum_{i=1}^NX_i}\right){\geq}\frac{1}{\mathbb{E}\left (\sum_{i=1}^nX_i/n\right)}=\frac{1}{\mathbb{E}(X)}$, and the limitation of $a_n$ may be $\lim_{n\to\infty}a_n=\mathbb{E}\left (\lim_{n\to\infty}\frac{1}{\frac{\sum_{i=1}^nX_i}{n}}\right)=\mathbb{E}\left(\frac{1}{\mathbb{E}(X)}\right)=\frac{1}{\mathbb{E}(X)}$. The question is: what is the convergence speed of $a_n\to\frac{1}{\mathbb{E}(X)}$?
According to simulation, the convergence speed seems like $\mathscr{O}(\frac{1}{n})$, but I wonder why.
Notes: May be here means I'm not sure whether the order of expectation and limitation can be exchanged here. It seems like correct through simulation results.