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