\(C=\frac{1}{1+2}+\frac{1}{1+2+3}+\frac{1}{1+2+3+4}+...+\frac{1}{1+2+3+...+10}\)
\(C=\frac{1}{3}+\frac{1}{6}+\frac{1}{10}...+\frac{1}{55}\)
\(C=\frac{1}{2}.\left(\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+...+\frac{1}{55}\right)\)
\(C.\frac{1}{2}=\frac{1}{3}.\frac{1}{2}+\frac{1}{6}.\frac{1}{2}+\frac{1}{10}.\frac{1}{2}+...+\frac{1}{55}.\frac{1}{2}\)
\(C.\frac{1}{2}=\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+...+\frac{1}{110}\)
\(C.\frac{1}{2}=\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{10.11}\)
\(C.\frac{1}{2}=\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{10}-\frac{1}{11}\)
\(C.\frac{1}{2}=\frac{1}{2}-\frac{1}{11}=\frac{11}{22}-\frac{2}{22}\)
\(C=\frac{9}{22}:\frac{1}{2}=\frac{9}{11}\)
Vậy \(C=\frac{9}{11}\)
\(C=\frac{1}{1+2}+\frac{1}{1+2+3}+\frac{1}{1+2+3+4}+...+\frac{1}{1+2+3+4+...+10}\)
\(C=\frac{1}{2.\left(2+1\right):2}+\frac{1}{3.\left(3+1\right):2}+\frac{1}{4.\left(4+1\right):2}+...+\frac{1}{10.\left(10+1\right):2}\)
\(C=\frac{2}{2.\left(2+1\right)}+\frac{2}{3.\left(3+1\right)}+\frac{2}{4.\left(4+1\right)}+...+\frac{2}{10.\left(10+1\right)}\)
\(C:2=\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{10.11}\)
\(C:2=\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{10}-\frac{1}{11}\)
\(C:2=\frac{1}{2}-\frac{1}{11}\)
\(C:2=\frac{9}{22}\)
\(C=\frac{9}{22}.2=\frac{9}{11}\)