Ta có A = 1/2^2 + 1/3^2 + 1/4^2 + ... + 1/100^2 = 1/2.2 + 1/3.3 + 1/4.4 + ... + 1/100.100 < 1/4 + 1/2.3 + 1/3.4 + ... + 1/99.100 A < 1/4 + 1/2 - 1/3 + 1/3 - 1/4 + ... + 1/99 - 1/100 = 1/4 + 1/2 - 1/100 = 3/4 - 1/100 \(\Rightarrow\) A < 3/4 ( đpcm )
Ta có: \(\frac{3}{4}=\frac{1}{4}+\frac{1}{2}\)
\(\Rightarrow A=\frac{1}{2^2}+\left(\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{100^2}\right)\)
\(\Rightarrow A< \frac{1}{2^2}+\left(\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+...+\frac{1}{99\cdot100}\right)\)
\(\Rightarrow A< \frac{1}{2^2}+\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{99}-\frac{1}{100}\right)\)
\(\Rightarrow A< \frac{1}{2^2}+\left(\frac{1}{2}-\frac{1}{100}\right)\)
\(\Rightarrow A< \frac{1}{2^2}+\frac{49}{100}\)
Mà \(\frac{49}{100}< \frac{1}{2}\)
\(\Rightarrow A< \frac{3}{4}\)
~Mik cũng hok Toán 2 ney~
