\(S=1+\frac{1}{3}+\frac{1}{6}+...+\frac{1}{45}\)
\(S=2.\left(\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+...+\frac{1}{90}\right)\)
\(S=2.\left(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{9.10}\right)\)
\(S=2.\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{9}-\frac{1}{10}\right)\)
\(S=2.\left(1-\frac{1}{10}\right)\)
\(S=2.\frac{9}{10}\)
\(S=\frac{9}{5}\)