\(P=\dfrac{1}{2^2}+\dfrac{1}{2^4}+\dfrac{1}{2^6}+...+\dfrac{1}{2^{2022}}\)
\(\Rightarrow\dfrac{8}{2}P=\dfrac{8}{2}\cdot\left(\dfrac{1}{2^2}+\dfrac{1}{2^4}+...+\dfrac{1}{2^{2022}}\right)\)
\(\Rightarrow4P=1+\dfrac{1}{2^2}+\dfrac{1}{2^4}+\dfrac{1}{2^6}+...+\dfrac{1}{2^{2020}}\)
\(\Rightarrow4P-P=\left(1+\dfrac{1}{2^2}+\dfrac{1}{2^4}+...+\dfrac{1}{2^{2020}}\right)-\left(\dfrac{1}{2^2}+\dfrac{1}{2^4}+...+\dfrac{1}{2^{2022}}\right)\)
\(\Rightarrow3P=\left(\dfrac{1}{2^2}-\dfrac{1}{2^2}\right)+\left(\dfrac{1}{2^4}-\dfrac{1}{2^4}\right)+...+\left(1-\dfrac{1}{2^{2022}}\right)\)
\(\Rightarrow3P=1-\dfrac{1}{2^{2022}}\)
\(\Rightarrow P=\dfrac{1-\dfrac{1}{2^{2022}}}{3}\)
Mà: \(1-\dfrac{1}{2^{2022}}< 1\)
\(\Rightarrow\dfrac{1-\dfrac{1}{2^{2022}}}{3}< \dfrac{1}{3}\)
\(\Rightarrow P< \dfrac{1}{3}\)