Expected value of the reciprocal

Question

Suppose $X_n>0$ is a sequence of positive random variables such that

$$ \mathbb{E}\left[X_n\right]\longrightarrow \infty\quad(1) $$

Given this, I am wondering whether there exist conditions to guarantee that

$$ \mathbb{E}\left[\frac{1}{X_n}\right]\longrightarrow 0. $$

or is it an immediate consequence of $(1)$?

Answer 1

Let $X_n$ be a positive random variable with: $$\mathsf P(X_n=n)=\frac12=\mathsf P(X_n=\frac1n)$$

Then $X_n$ and $X_n^{-1}$ have the same distribution.

This with: $$\mathsf EX_n^{-1}=\mathsf EX_n=\frac12n+\frac12\frac1n\to\infty$$

I have no direct answer on your request for sufficient conditions.