M=1/2^2+1/3^2+1/4^2+...+1/9^2.CM 2/5<M<1

Đáp án+Giải thích các bước giải:

$M=\dfrac{1}{2^2}+\dfrac{1}{3^2}+\dfrac{1}{4^2}+\dots+\dfrac{1}{9^2}\\ =\dfrac{1}{2.2}+\dfrac{1}{3.3}+\dfrac{1}{4.4}+\dots+\dfrac{1}{9.9}\\ <\dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+\dots+\dfrac{1}{8.9}\\ =1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+\dots+\dfrac{1}{8}-\dfrac{1}{9}\\ =1-\dfrac{1}{9}\\ <1\\ \Rightarrow M<1(*)\\ M=\dfrac{1}{2^2}+\dfrac{1}{3^2}+\dfrac{1}{4^2}+\dots+\dfrac{1}{9^2}\\ =\dfrac{1}{2.2}+\dfrac{1}{3.3}+\dfrac{1}{4.4}+\dots+\dfrac{1}{9.9}\\ >\dfrac{1}{2.3}+\dfrac{1}{3.4}+\dfrac{1}{4.5}+\dots+\dfrac{1}{9.10}\\ =\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{5}+\dots+\dfrac{1}{9}-\dfrac{1}{10}\\ =\dfrac{1}{2}-\dfrac{1}{10}\\ =\dfrac{2}{5}\\ \Rightarrow M>\dfrac{2}{5} (**)\\ (*)(**) \Rightarrow \dfrac{2}{5}<M<1.$