`A = 1/(2^2) + 1/(3^2) + 1/(4^2) + ... + 1/(100^2)`
`A = 1 - (1 - 1/2) + (1 - 1/3) + (1 - 1/4) + ... + (1 - 1/100)`
`=> A < 1`
Bổ sung thêm:
`A = 1 - 1/1 + 2/3 + 3/4 + ... + 99/100`
Ta có: \(\dfrac{1}{2^2}< \dfrac{1}{2}\)
\(\dfrac{1}{3^2}< \dfrac{1}{2.3}\)
\(\dfrac{1}{4^2}< \dfrac{1}{3.4}\)
...
\(\dfrac{1}{100^2}< \dfrac{1}{99.100}\)
Cộng vế với vế, ta có:
\(A=\dfrac{1}{2^2}+\dfrac{1}{3^2}+\dfrac{1}{4^2}+...+\dfrac{1}{100^2}< \dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+...+\dfrac{1}{99.100}\)
Lại có:
\(\dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+...+\dfrac{1}{99.100}=\dfrac{2-1}{1.2}+\dfrac{3-2}{2.3}+\dfrac{4-3}{3.4}+...+\dfrac{100-99}{99.100}\)
\(=1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{99}-\dfrac{1}{100}\)
\(=1-\dfrac{1}{100}=\dfrac{99}{100}< 1\)
\(\Rightarrow A< 1\)
