Let $(a_n)_{n \geq 1}$ be a nonnegative, strictly decreasing sequence with $a_n \to 0$ as $n \to \infty$.

Consider the ratio (for $k \geq n$) $$ R_{n, k} = \mathbb{E}_{u \sim \text{Unif}(\mathbb{S}^{k-1})} \Big[\frac{a_n u_n^2}{\sum_{m \leq n} a_m u_m^2} \Big]. $$ Is it possible to compute the limit $$ R^\star_n = \lim_{k \to \infty} R_{n, k}? $$

Let $(a_n)_{n \geq 1}$ be a nonnegative, strictly decreasing sequence with $a_n \to 0$ as $n \to \infty$.

Consider the ratio (for $k \geq n$) $$ R_{n, k} = \mathbb{E}_{u \sim \text{Unif}(\mathbb{S}^{k-1})} \Big[\frac{a_n u_n^2}{\sum_{m=1}^k a_m u_m^2} \Big]. $$ Is it possible to compute the limit $$ R^\star_n = \lim_{k \to \infty} R_{n, k}? $$