Question

A= 1/3² +1/4² +1/5² +……1/100 chứng minh A<1/2

Answer 1

Ta có :

\(\dfrac{1}{3^2}=\dfrac{1}{3\times3}< \dfrac{1}{2\times3}\)

\(\dfrac{1}{4^2}=\dfrac{1}{4\times4}< \dfrac{1}{3\times4}\)

Tương tự như vậy ta cũng được \(\dfrac{1}{100}=\dfrac{1}{10\times10}< \dfrac{1}{9\times10}\)

Do đó :

\(A< \dfrac{1}{2\times3}+\dfrac{1}{3\times4}+.....+\dfrac{1}{9\times10}\)

Suy ra \(A< \dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+.....+\dfrac{1}{9}-\dfrac{1}{10}=\dfrac{1}{2}-\dfrac{1}{10}=\dfrac{2}{5}< \dfrac{1}{2}\)

Do đó \(A< \dfrac{1}{2}\left(đpcm\right)\)