Question

[Mai Tuấn Kiệt]

Chứng minh rằng :\(\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+............+\frac{1}{n^2}< 2-\frac{1}{n}\)

Answer 1

[Kuroba Kaito]

Đặt A =  \(\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{n^2}\)< \(1+\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{\left(n-1\right).n}\)

=> A < 1 + (1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + ... + 1/(n - 1) - 1/n)

=> A < 1 + (1 - 1/n)

=> A < 2 - 1/n