\(R\left(x\right)=x^2-x=x\left(x-1\right)\)
\(\Leftrightarrow S=\dfrac{1}{3\left(3-1\right)}+\dfrac{1}{4\left(4-1\right)}+\dfrac{1}{5\left(5-1\right)}+...+\dfrac{1}{2023\left(2023-1\right)}+\dfrac{1}{2.2023}\)
\(\Leftrightarrow S=\dfrac{1}{3.2}+\dfrac{1}{4.3}+\dfrac{1}{5.4}+...+\dfrac{1}{2023.2022}+\dfrac{1}{4046}\)
\(\Leftrightarrow S=\dfrac{3-2}{3.2}+\dfrac{4-3}{4.3}+\dfrac{5-4}{5.4}+...+\dfrac{2023-2022}{2023.2022}+\dfrac{1}{4046}\)
\(\Leftrightarrow S=\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{5}+...+\dfrac{1}{2022}-\dfrac{1}{2023}+\dfrac{1}{4046}=\dfrac{1}{2}-\dfrac{1}{2023}+\dfrac{1}{4046}=\dfrac{2023-2+1}{4046}=\dfrac{2022}{4046}=\dfrac{1011}{2023}\)
