$\begin{align}
& 1-\frac{5}{6}+\frac{7}{12}-\frac{9}{20}+\frac{11}{30}-\frac{13}{42}+\frac{15}{56}-\frac{17}{72}+\frac{19}{90} \\
& =1-\frac{5}{2\times 3}+\frac{7}{3\times 4}-\frac{9}{4\times 5}+\frac{11}{5\times 6}-\frac{13}{6\times 7}+\frac{15}{7\times 8}-\frac{17}{8\times 9}+\frac{10}{9\times 10} \\
& =1-\frac{3+2}{2\times 3}+\frac{4+3}{3\times 4}-\frac{5+4}{4\times 5}+\frac{6+5}{5\times 6}-\frac{7+6}{6\times 7}+\frac{8+7}{7\times 8}-\frac{9+8}{8\times 9}+\frac{10+9}{9\times 10} \\
& =1-\frac{1}{2}-\frac{1}{3}+\frac{1}{3}+\frac{1}{4}-\frac{1}{4}-\frac{1}{5}+\frac{1}{5}+\frac{1}{6}-\frac{1}{6}-\frac{1}{7}+\frac{1}{7}+\frac{1}{8}-\frac{1}{8}-\frac{1}{9}+\frac{1}{9}+\frac{1}{10} \\
& =1-\frac{1}{2}+\frac{1}{10}=\frac{1}{2}+\frac{1}{10}=\frac{6}{10}=\frac{3}{5} \\
\end{align}$

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