\(\frac12+\frac16+\frac{1}{12}+\frac{1}{20}+\cdots+\frac{1}{90}\)
\(=\frac{1}{1*2}+\frac{1}{2*3}+\frac{1}{3*4}+\frac{1}{4*5}+\cdots+\frac{1}{9*10}\)
\(=1-\frac12+\frac12-\frac13+\frac13-\frac14+\frac14-\cdots+\frac19-\frac{1}{10}\)
\(=1-\frac{1}{10}=\frac{9}{10}\)