\(\frac{1}{1+2+3}+\frac{1}{1+2+3+4}+...+\frac{1}{1+2+3+...+59}\)
\(=\frac{1}{\frac{3.4}{2}}+\frac{1}{\frac{4.5}{2}}+...+\frac{1}{\frac{59.60}{2}}\)
\(=\frac{2}{3.4}+\frac{2}{4.5}+...+\frac{2}{59.60}\)
\(=2.\left(\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{59.60}\right)\)
\(=2.\left(\frac{4-3}{3.4}+\frac{5-4}{4.5}+...+\frac{60-59}{59.60}\right)\)
\(=2.\left(\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{59}-\frac{1}{60}\right)\)
\(=2.\left(\frac{1}{3}-\frac{1}{60}\right)\)
\(=2.\frac{19}{60}\)
\(=\frac{38}{60}\)\(< \frac{40}{60}=\frac{2}{3}\)
