Question

[Người Ấy Là Ai]

CMR :1/1^2 + 1/2^2 + 1/3^2 + ... + 1/n^2 < 5/3

Answer 1

[Ánh Lê]

Ta có \(a^2>a^2-1\forall a\)

\(\Rightarrow a^2>\left(a-1\right)\left(a+1\right)\)

\(\Rightarrow\dfrac{1}{a^2}< \dfrac{1}{\left(a-1\right)\left(a+1\right)}=\dfrac{1}{2}\cdot\left(\dfrac{1}{a-1}\right)\left(\dfrac{1}{a+1}\right)\)

Áp dụng, ta có

\(\dfrac{1}{1^2}+\dfrac{1}{2^2}+\dfrac{1}{3^2}+\dfrac{1}{4^2}+...+\dfrac{1}{n^2}< 1+\dfrac{1}{2^2}+\dfrac{1}{2\cdot4}+\dfrac{1}{3\cdot5}+...+\dfrac{1}{\left(n-1\right)\left(n+1\right)}\)

= \(1+\dfrac{1}{2^2}+\dfrac{1}{2}\cdot\left(\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{3}-\dfrac{1}{5}+...+\dfrac{1}{n-1}-\dfrac{1}{n+1}\right)\)

= 1+ \(\dfrac{1}{4}\)+\(\dfrac{1}{2}\cdot\left(\dfrac{1}{2}+\dfrac{1}{3}-\dfrac{1}{n}-\dfrac{1}{n+1}\right)\)

=1+ \(\dfrac{2}{3}-\dfrac{1}{2}\cdot\left(\dfrac{1}{n}+\dfrac{1}{n+1}\right)\) < \(1+\dfrac{2}{3}=\dfrac{5}{3}\left(ĐPCM\right)\)

Answer 2

[Phùng Tuệ Minh]

(Mik mượn chỗ bình luận ké nha!!)

Người Ấy Là Ai-eqt đẹp đó :)