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Question Number 186453 by norboyev last updated on 04/Feb/23

$$\frac{1}{1+2}+\frac{1}{1+2+3}+\frac{1}{1+2+3+4}+...+\frac{1}{1+2+3+...+10}=?$$

Answered by ARUNG_Brandon_MBU last updated on 04/Feb/23

$$S=\frac{1}{1+2}+\frac{1}{1+2+3}+\frac{1}{1+2+3+4}+\cdot \cdot \cdot +\frac{1}{1+2+3+\cdot \cdot \cdot +10}$$$$=\underset{n=2}{\sum ^{10}}\frac{1}{\underset{k=1}{\sum ^n}k}=\underset{n=2}{\sum ^{10}}\frac{2}{n(n+1)}=2\underset{n=2}{\sum ^{10}}(\frac{1}{n}-\frac{1}{n+1})$$$$=2((\frac{1}{2}-\frac{1}{3})+(\frac{1}{3}-\frac{1}{4})+\cdot \cdot \cdot +(\frac{1}{10}-\frac{1}{11}))$$$$=2(\frac{1}{2}-\frac{1}{11})=2\left(\frac{11-2}{22}\right)=\frac{9}{11}$$

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