Question

[Kinomoto Sakura]

A =1/2²+1/3²+1/4²+...+1/n²

chứng minh A<2/3

Answer 1

[IR IRAN(Islamic Republic of Iran)]

Ta có : A =\(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{n^2}=\frac{1}{2.2}+\frac{1}{3.3}+\frac{1}{4.4}+...+\frac{1}{n.n}\)

< \(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{\left(n-1\right)n}\)

= \(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{n-1}-\frac{1}{n}\)

= \(1-\frac{1}{n}=\frac{n-1}{n}\)

Ta có 3(n - 1) = 3n - 3

2n = 2n

Mà 3n - 3 > 2n

=> 3(n - 1) > 2n

=> \(\frac{2}{3}>\frac{n-1}{n}\)(tính chất tỉ lệ thức)

<=> A < 2/3 (ĐPCM)