Question 136
Choose positive integers $b_1, b_2, \dots$ satisfying
$$
1 = \frac{b_1}{1^2} \gt \frac{b_2}{2^2} \gt \frac{b_3}{3^2} \gt \frac{b_4}{4^2} \gt \dots
$$
and let $r$ denote the largest real number satisfying $\frac{b_n}{n^2} \ge r$ for all positive integers $n$. What are the possible values of $r$ across all possible choices of the sequence $(b_n)$?
$$
1 = \frac{b_1}{1^2} \gt \frac{b_2}{2^2} \gt \frac{b_3}{3^2} \gt \frac{b_4}{4^2} \gt \dots
$$
and let $r$ denote the largest real number satisfying $\frac{b_n}{n^2} \ge r$ for all positive integers $n$. What are the possible values of $r$ across all possible choices of the sequence $(b_n)$?