Verified answer

Ответ:

0.505

Объяснение:

[tex]$ \left(1 - \cfrac{1}{4}\right) \left(1 - \cfrac{1}{9}\right) \left(1 - \cfrac{1}{16}\right) \dots \left(1 - \cfrac{1}{100^2}\right) = \left(1 - \cfrac{1}{2^2}\right) \left(1 - \cfrac{1}{3^2}\right) \left(1 - \cfrac{1}{4^2}\right) \dots \left(1 - \cfrac{1}{100^2}\right) =[/tex]

[tex]= $\left(\cfrac{2^2 - 1}{2^2}\right) \left(\cfrac{3^2 - 1}{3^2}\right) \left(\cfrac{4^2 - 1}{4^2}\right) \dots \left(\cfrac{100^2 - 1}{100^2}\right) =[/tex]

=[tex]\cfrac{(2^2 - 1)*(3^2 - 1)*(4^2 - 1) *\dots* (100^2 - 1)}{2^2*3^2*4^2*\dots* 100^2}[/tex]

[tex]\cfrac{(2 - 1)(2 + 1)*(3 - 1)(3 + 1)*(4 - 1)(4 + 1) *\dots* (100 - 1)(100 + 1)}{2^2*3^2*4^2*\dots* 100^2} =[/tex]

[tex]\cfrac{1*3*2*4*3*5 *\dots*97*99*98 * 100 * 99* 101}{2^2*3^2*4^2*\dots* 100^2} = \cfrac{1*2*3^2*4^2*5^2 *\dots* 99^2* 100 * 101}{2^2*3^2*4^2*\dots* 100^2} =[/tex]

[tex]=\cfrac{101}{2* 100} = \cfrac{101}{200} = 0.505[/tex]

101/200

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