VTRMC 2018 Problem 6

For $n \in \mathbb{N}$ , define $a_n = \frac{1 + 1/3 + 1/5 + \dots + 1/(2n-1)}{n+1}$ and $b_n = \frac{1/2 + 1/4 + 1/6 + \dots + 1/(2n)}{n}$ . Find the maximum and minimum of $a_n - b_n$ for $1 \leq n \leq 999$ .