\(S=\dfrac{1}{2}+\dfrac{9}{22}+\dfrac{7}{198}+\dfrac{5}{414}\)
\(=\dfrac{1}{2}+\dfrac{9}{2\cdot11}+\dfrac{7}{11\cdot18}+\dfrac{5}{18\cdot23}\)
\(=\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{11}+\dfrac{1}{11}-\dfrac{1}{18}+\dfrac{1}{18}-\dfrac{1}{23}\)
\(=1-\dfrac{1}{23}=\dfrac{22}{23}\)