Using factorization to solve a problem (1-1/2^2 )(1-1/3^2 )(1-1/4^2 )^…… (1-1 / N ^ 2) use factorization to solve this problem

Using factorization to solve a problem (1-1/2^2 )(1-1/3^2 )(1-1/4^2 )^…… (1-1 / N ^ 2) use factorization to solve this problem

First of all, let's look at the general term 1-1 / N ^ 2 = (1-1 / N) (1 + 1 / N), so (1-1 / 2 ^ 2) (1-1 / 3 ^ 2) (1-1 / 4 ^ 2) ^ (1-1/n^2 ) =(1-1/2)(1+1/2)(1-1/3)(1+1/3).(1-1/n)(1+1/n) =【(1-1/2)(1-1/3).(1-1/n)】【(1+1/2)(1+1/3).(1+1/n)】 =【1/2×2/3×.×(n-1)/n】【3/2×4/3×.×(n+1)/n】 = 【1 / n 】× 【(n+1)/2】 =(n+1)/2n