A = \(\dfrac{3}{2}-\dfrac{5}{6}+\dfrac{7}{12}-\dfrac{9}{20}+\dfrac{11}{30}-\dfrac{13}{42}+\dfrac{15}{56}-\dfrac{17}{72}+\dfrac{19}{90}\)
A \(=\dfrac{3}{1\cdot2}-\dfrac{5}{2\cdot3}+\dfrac{7}{3\cdot4}-\dfrac{9}{4\cdot5}+\dfrac{11}{5\cdot6}-\dfrac{13}{6\cdot7}+\dfrac{15}{7\cdot8}-\dfrac{17}{8\cdot9}+\dfrac{19}{9\cdot10}\)
A
\(=\left(1+\dfrac{1}{2}\right)+\left(\dfrac{1}{2}+\dfrac{2}{3}\right)+\left(\dfrac{1}{3}+\dfrac{3}{4}\right)+\left(\dfrac{1}{4}+\dfrac{1}{5}\right)+\left(\dfrac{1}{5}+\dfrac{1}{6}\right)+\left(\dfrac{1}{6}+\dfrac{1}{7}\right)+\left(\dfrac{1}{7}+\dfrac{7}{8}\right)+\left(\dfrac{1}{8}+\dfrac{1}{9}\right)+\left(\dfrac{1}{9}+\dfrac{1}{10}\right)\)
\(A=1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{8}+\dfrac{1}{8}-\dfrac{1}{9}+\dfrac{1}{9}-\dfrac{1}{10}\)
A = \(1-\dfrac{1}{10}=\dfrac{9}{10}\)
