Đặt \(S=\dfrac{1}{2^2}+\dfrac{1}{3^2}+\dfrac{1}{4^2}+...+\dfrac{1}{10^2}\)
Ta có:
\(S< \dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+...+\dfrac{1}{9.10}\)
\(\Rightarrow S< \dfrac{1}{1}-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{9}-\dfrac{1}{10}\)
\(\Rightarrow S< 1-\dfrac{1}{10}< 1\) (1)
Lại có:
\(S>\dfrac{1}{2.3}+\dfrac{1}{3.4}+\dfrac{1}{4.5}+...+\dfrac{1}{10.11}\)
\(\Rightarrow S>\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{10}-\dfrac{1}{11}\)
\(\Rightarrow S>\dfrac{1}{2}-\dfrac{1}{11}=\dfrac{9}{22}>\dfrac{8}{22}=\dfrac{4}{11}\) (2)
Từ (1);(2) \(\Rightarrow\dfrac{4}{11}< S< 1\)
