I must figure out $\lim_{k \to \infty} \mathbb{E}\left[ W_k \right]$ for $$ W_k = \frac{1}{a+b \cdot\chi^2_k} \qquad a,b \in \mathbb{R}_+$$ where $\chi^2_k $ indicates a chi-squared random variable with $k$ degrees of freedom. I know that $X = \lim_{n \to \infty} \chi^2_k \sim \mathcal{N}(0,1)$ but in the limit $\mathbb{E}\left[ \frac{1}{X} \right]$ does not converge. Any idea of how to obtain the limit of the expectation $\lim_{k \to \infty} \mathbb{E}\left[ W_k \right]$ analytically?
Many thanks
Let $(Y_i)_{i\geqslant 1}$ be a sequence of independent standard normal random variables. We know that for each $k$, $W_k$ has the same distribution as $$ W'_k=\frac{1}{a+b\sum_{i=1}^kY_i^2} $$ hence we are looking for $$ \lim_{k\to\infty}\mathbb E\left[\frac{1}{a+b\sum_{i=1}^kY_i^2}\right]. $$ We can apply the reversed monotone convergence theorem, since $(W'_k)_{k\geqslant 1}$ is non-increasing and $W'_1$ is integrable. The strong law of large numbers shows that $W'_k\to 0$ almost surely hence $\mathbb E[W_n]\to 0$.