Question

[nguyen tuan vinh]

Cho A= 1/2^2+1/3^2+1/4^2+...+1/n^2

CMR:A<1

Answer 1

[Lovely Sweetheart Princess]

A= \(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{n^2}< \frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{\left(n-1\right)n}\)

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

A<\(1-\frac{1}{n}\)=\(\frac{n}{n}-\frac{1}{n}=\frac{n-1}{n}< 1\)

Vậy A < 1

Answer 2

[Vũ Quang Vinh]

Ta có:
1/2² < 1/1.2
1/3² < 1/2.3
1/4² < 1/3.4
..................
=> 1/n² < 1/n(n-1)
=> 1/2² + 1/3² + 1/4² + ... + 1/n² < 1/1.2 + 1/2.3 + 1/3.4 + ... + 1/n(n-1)
=> A < 1/1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + ... + 1/n-1 + 1/n
=> A < 1 - 1/n
Vơi n thuộc N* => 1 - 1/n < 1 ( vì 1/n lúc đó lớn hơn 0 )
=> A < 1 - 1/n < 1
đpcm