Câu hỏi của Mai Tuấn Kiệt

Question

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

Kuroba Kaito · Accepted Answer

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

Câu trả lời:

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