Prove inequality: (2-1) / (2 ^ 2-1) + (2 ^ 2-1) / (2 ^ 3-1) +. + (2 ^ n-1) / (2 ^ n + 1) - 1) > n / 2-1 / 3

Prove inequality: (2-1) / (2 ^ 2-1) + (2 ^ 2-1) / (2 ^ 3-1) +. + (2 ^ n-1) / (2 ^ n + 1) - 1) > n / 2-1 / 3

The original inequality is equivalent to (2 ^ 2-2) / (2 ^ 2-1) + (2 ^ 3-2) / (2 ^ 3-1) +... + (2 ^ (n + 1) - 2) / (2 ^ (n + 1) - 1) > n-2 / 3
Left end = (1-1 / (2 ^ 2-1)) + (1-1 / (2 ^ 3-1)) +... (1-1 / (2 ^ (n + 1) - 1)) = n - (1 / (2 ^ 2-1) + 1 / (2 ^ 3-1) +... + 1 / (2 ^ (n + 1) - 1))
So the inequality is further equivalent to 1 / (2 ^ 2-1) + 1 / (2 ^ 3-1) +... + 1 / (2 ^ (n + 1) - 1) < 2 / 3
When k > 2, there is 2 ^ (K-2) > 1, so 2 ^ k-1 > 2 ^ K-2 ^ (K-2) = 3.2 ^ (K-2)
So 1 / (2 ^ 2-1) + 1 / (2 ^ 3-1) +... + 1 / (2 ^ (n + 1) - 1)
< 1/3+1/(3·2)+...+1/(3·2^(n-1))
= 2/3-1/(3·2^(n-1))
< 2/3.