Bonsoir,
[tex]P=(1-(\frac{1}{2} )^2)*(1-(\frac{1}{3} )^2)*(1-(\frac{1}{4} )^2)*....*(1-(\frac{1}{100} )^2)\\\\1-(\frac{1}{2} )^2=(1-\frac{1}{2} )*(1+\frac{1}{2} )=\frac{1}{2} *\frac{3}{2} \\\\1-(\frac{1}{3} )^2=(1-\frac{1}{3} )*(1+\frac{1}{3} )=\frac{2}{3} *\frac{4}{3} \\\\1-(\frac{1}{4} )^2=(1-\frac{1}{4} )*(1+\frac{1}{4} )=\frac{3}{4} *\frac{5}{4} \\\\...\\1-(\frac{1}{n} )^2=(1-\frac{1}{n} )*(1+\frac{1}{n} )=\frac{n-1}{n} *\frac{n+1}{n} \\\\[/tex]
[tex](1-(\frac{1}{2} )^2)*(1-(\frac{1}{3} )^2)*(1-(\frac{1}{4} )^2)*....*\frac{n-1}{n} *\frac{n+1}{n}\\\\=[\frac{1}{2}*\frac{3}{2}]*[\frac{2}{3}*\frac{4}{3}]*[\frac{3}{4}*\frac{5}{4}]*...*[\frac{n-1}{n}*\frac{n+1}{n}]\\\\=\frac{1}{2}*\frac{n+1}{n}\\\\si\ n=100\ alors\ \\\\ \displaystyle \prod_{k=2}^{k=100} (1-\frac{1}{k^2})=\frac{1}{2}*\frac{101}{100}=\frac{101}{200}[/tex]
