Question

[Lý Nguyễn Huy Hoàng]

6) Chứng minh rằng : A = 1/2² + 1/4² + 1/6² + 1/8² + 1/10² + ... + 1/50² < 1/2 

SOS !!! HELP ME !!!

Answer 1

[Nguyễn Lê Phước Thịnh]

\(\dfrac{1}{2^2}< \dfrac{1}{1\cdot2}=1-\dfrac{1}{2}\)

\(\dfrac{1}{3^2}< \dfrac{1}{2\cdot3}=\dfrac{1}{2}-\dfrac{1}{3}\)

...

\(\dfrac{1}{25^2}< \dfrac{1}{24\cdot25}=\dfrac{1}{24}-\dfrac{1}{25}\)

Do đó: \(\dfrac{1}{2^2}+\dfrac{1}{3^2}+...+\dfrac{1}{25^2}< 1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{24}-\dfrac{1}{25}\)

=>\(\dfrac{1}{2^2}+\dfrac{1}{3^2}+...+\dfrac{1}{25^2}< 1-\dfrac{1}{25}\)

=>\(1+\dfrac{1}{2^2}+...+\dfrac{1}{25^2}< 2-\dfrac{1}{25}\)

=>\(A=\dfrac{1}{2^2}\left(1+\dfrac{1}{2^2}+...+\dfrac{1}{25^2}\right)< \dfrac{1}{4}\left(2-\dfrac{1}{25}\right)=\dfrac{1}{2}-\dfrac{1}{100}< \dfrac{1}{2}\)