Bộ ông rảnh rỗi sinh nông nổi ak ??
Ta có :
\(A=\dfrac{1}{3^2}+\dfrac{1}{6^2}+\dfrac{1}{9^2}+....................+\dfrac{1}{9n^2}\)
\(\Rightarrow A=\dfrac{1}{\left(3.1\right)^2}+\dfrac{1}{\left(3.2\right)^2}+\dfrac{1}{\left(3.3\right)^2}+...................+\dfrac{1}{\left(3n\right)^2}\)
\(\Rightarrow A=\dfrac{2}{9}\left(\dfrac{1}{1^2}+\dfrac{1}{2^2}+\dfrac{1}{3^2}+..............+\dfrac{1}{n^2}\right)\)
\(\Rightarrow A< \dfrac{2}{9}\left(\dfrac{1}{1}+\dfrac{1}{1.2}+\dfrac{1}{2.3}+..................+\dfrac{1}{\left(n-1\right)n}\right)\)
\(\Rightarrow A< \dfrac{2}{9}\left(1+1-\dfrac{1}{2}+\dfrac{1}{3}-\dfrac{1}{4}+.........+\dfrac{1}{n-1}-\dfrac{1}{n}\right)\)
\(\Rightarrow A< \dfrac{2}{9}\left(1+1-\dfrac{1}{n}\right)\)
\(\Rightarrow A< \dfrac{2}{9}\left(2-\dfrac{1}{n}\right)< \dfrac{2}{9}\)
\(\Rightarrow A< \dfrac{2}{9}\rightarrowđpcm\)
P/S : Lâu lâu ko ôn dạng này nên quên hết ồi!!
