Đáp án:
`A<B`
Giải thích các bước giải:
Xét `A`
`⇒A=(1-\frac{1}{2})(1-\frac{1}{3})(1-\frac{1}{4})....(1-\frac{1}{19})(1-\frac{1}{20})`
`⇒A=\frac{1}{2}.\frac{2}{3}.\frac{3}{4}.....\frac{18}{19}.\frac{19}{20}`
`⇒A=\frac{1.(2.3......19)}{(2.3.4.....19).20}`
`⇒A=\frac{1}{20}<\frac{1}{2}(1)`
Xét `B`
`⇒B=(1-\frac{1}{4})(1-\frac{1}{9})(1-\frac{1}{16})...(1-\frac{1}{100^2})`
`⇒B=\frac{3}{2^2}.\frac{8}{3^2}.\frac{15}{4^2}....\frac{9999}{100^2}`
`⇒B=\frac{1.3}{2^2}.\frac{2.4}{3^2}.\frac{3.5}{4^2}....\frac{99.101}{100^2}`
`⇒B=\frac{1.2.3...98.99}{2.3.4...99.100}.\frac{3.4.5...100.101}{2.3.4...100}`
`⇒B=\frac{1}{100}.\frac{101}{2}=\frac{101}{200}>\frac{1}{2}(2)`
Từ `(1)(2)⇒A<B`
