\(=\dfrac{1}{11}+\dfrac{1}{11\cdot19}+\dfrac{1}{19\cdot27}+\dfrac{1}{27\cdot35}+\dfrac{1}{35\cdot43}+\dfrac{1}{43\cdot51}\)
\(=\dfrac{1}{11}+\dfrac{1}{8}\left(\dfrac{1}{11}-\dfrac{1}{19}+\dfrac{1}{19}-...-\dfrac{1}{51}\right)\)
\(=\dfrac{1}{11}+\dfrac{1}{8}\cdot\dfrac{40}{51\cdot11}\)
\(=\dfrac{1}{11}+\dfrac{5}{51\cdot11}=\dfrac{56}{51\cdot11}\)
\(\dfrac{1}{11}+\dfrac{1}{209}+\dfrac{1}{513}+\dfrac{1}{945}+\dfrac{1}{1505}+\dfrac{1}{2193}\)
\(=\dfrac{1}{11}+\dfrac{1}{11.19}+\dfrac{1}{19.27}+...+\dfrac{1}{43.51}\)
\(=\dfrac{1}{11}+\dfrac{1}{8}\left(\dfrac{8}{11.19}+\dfrac{8}{19.27}+...+\dfrac{8}{43.51}\right)\)
\(=\dfrac{1}{11}+\dfrac{1}{8}\left(\dfrac{1}{11}-\dfrac{1}{19}+\dfrac{1}{19}-\dfrac{1}{27}+...+\dfrac{1}{43}-\dfrac{1}{51}\right)\)
\(=\dfrac{1}{11}+\dfrac{1}{8}\left(\dfrac{1}{11}-\dfrac{1}{51}\right)\)
\(=\dfrac{1}{11}+\dfrac{5}{561}=\dfrac{56}{561}\)
