Giải thích các bước giải:
$A=\dfrac{1}{2^2}+\dfrac{1}{3^2}+\cdots+\dfrac{1}{1005^2}\\ =\dfrac{1}{2.2}+\dfrac{1}{3.3}+\cdots+\dfrac{1}{1005.1005}\\ <\dfrac{1}{1.2}+\dfrac{1}{2.3}+\cdots+\dfrac{1}{1004.1005}$
Ta có:
$\dfrac{1}{1.2}=\dfrac{1}{1}-\dfrac{1}{2}\\ \dfrac{1}{2.3}=\dfrac{1}{2}-\dfrac{1}{3}\\ \cdots\\ \dfrac{1}{1004.1005}=\dfrac{1}{1004}-\dfrac{1}{1005}\\ \Rightarrow A<\dfrac{1}{1}-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\cdots+\dfrac{1}{1004}-\dfrac{1}{1005}\\ \Leftrightarrow A<1-\dfrac{1}{1005}\\ \Leftrightarrow A<\dfrac{1004}{1005}$