\(\left(1-\frac{1}{2^2}\right).\left(1-\frac{1}{3^2}\right).\left(1-\frac{1}{4^2}\right)...\left(1-\frac{1}{n^2}\right)\)
\(=\frac{2^2-1}{2^2}.\frac{3^2-1}{3^2}.\frac{4^2-1}{4^2}...\frac{n^2-1}{n^2}\)
\(=\frac{\left(2-1\right).\left(2+1\right)}{2^2}.\frac{\left(3-1\right).\left(3+1\right)}{3^2}.\frac{\left(4-1\right).\left(4+1\right)}{4^2}...\frac{\left(n-1\right).\left(n+1\right)}{n^2}\)
\(=\frac{1.3}{2^2}.\frac{2.4}{3^2}.\frac{3.5}{4^2}...\frac{\left(n-1\right).\left(n+1\right)}{n^2}\)
\(=\frac{1.2.3...\left(n-1\right)}{2.3.4...n}.\frac{3.4.5...\left(n+1\right)}{2.3.4...n}\)
\(=\frac{1}{n}.\frac{n+1}{2}=\frac{\left(n+1\right)}{2n}\)
