Bài 5: So sánh A với -1/2, A=(1/2^2-1)(1/3^2-1)(1/4^2-1)...(1/100^2-1)

Minhdq

A là tích của 99 số âm nên ta có:

$-A=\left(1-\frac{1}{4}\right)\left(1-\frac{1}{9}\right)\left(1-\frac{1}{16}\right)\ldots\left(1-\frac{1}{100^2}\right)$

$=\frac{3}{2^2}.\frac{8}{3^2}.\frac{15}{4^2}\ldots\frac{9999}{100^2}$

$=\frac{1.3}{2^2}.\frac{2.4}{3^2}.\frac{3.5}{4^2}\ldots\frac{99.101}{100^2}$

$=\frac{1.2.3.4.5.6\ldots98.99}{2.3.4.5\ldots99.100}.\frac{3.4.5\ldots100.101}{2.3.4\ldots99.100}$

$=\frac{1}{100}.\frac{101}{2}$

$=\frac{101}{200}>\frac{1}{2}$

$\Rightarrow A<\frac{-1}{2}$.