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 
$$
R^\star_n = \lim_{k \to \infty} R_{n, k}?
$$