2 / 1 * 4 / 3 * 6 / 5 * 8 / 7 *. (2n) / (2n - 1) > √ (2n + 1)

2 / 1 * 4 / 3 * 6 / 5 * 8 / 7 *. (2n) / (2n - 1) > √ (2n + 1)


2 ^ 2 > 1 * 3
4 ^ 2 > 3 * 5
...
(2n) ^ 2 > (2n - 1) (2n + 1)
∴ (2 * 4 * 6 * 8 *... 2n) ^ 2 > 1 * 3 ^ 2 * 5 ^ 2 * 7 ^ 2 *.. (2n - 1) ^ 2 * (2n + 1)
∴ [(2 * 4 * 6 * 8 *... 2n) / (1 * 3 * 5 * 7 *... (2n - 1)] ^ 2 > 2n + 1
∴ 2 / 1 * 4 / 3 * 6 / 5 * 8 / 7 *. (2n) / (2n - 1) > √ (2n + 1)