\(T=\dfrac{1}{1\cdot2}+\dfrac{5}{2\cdot3}+\dfrac{11}{3\cdot4}+...+\dfrac{89}{9\cdot10}\)

\(T=\left(1-\dfrac{1}{1\cdot2}\right)+\left(1-\dfrac{1}{2\cdot3}\right)+\left(1-\dfrac{1}{3\cdot4}\right)+...+\left(1-\dfrac{1}{9\cdot10}\right)\)

\(T=9-\left(\dfrac{1}{1\cdot2}+\dfrac{1}{2\cdot3}+...+\dfrac{1}{9\cdot10}\right)\)

\(T=9-\left(\dfrac{1}{1}-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{9}-\dfrac{1}{10}\right)\)

\(T=9-\left(\dfrac{1}{1}-\dfrac{1}{10}\right)=9-\dfrac{9}{10}=\dfrac{81}{10}\)