Question Number 160948 by Rustambek last updated on 09/Dec/21
$$\frac{\mathrm{1}}{\mathrm{1}\centerdot\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{3}\centerdot\mathrm{4}}+\frac{\mathrm{1}}{\mathrm{5}\centerdot\mathrm{6}}+…+\frac{\mathrm{1}}{\mathrm{39}\centerdot\mathrm{40}}=? \\ $$
Answered by puissant last updated on 09/Dec/21
$$\Omega=\underset{{k}=\mathrm{1}} {\overset{\mathrm{39}} {\sum}}\frac{\mathrm{1}}{{k}\left({k}+\mathrm{1}\right)}\:=\:\underset{{k}=\mathrm{1}} {\overset{\mathrm{39}} {\sum}}\left\{\frac{\mathrm{1}}{{k}}−\frac{\mathrm{1}}{{k}+\mathrm{1}}\right\} \\ $$$$=\:\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{3}}+…….−\frac{\mathrm{1}}{\mathrm{39}}+\frac{\mathrm{1}}{\mathrm{39}}−\frac{\mathrm{1}}{\mathrm{40}} \\ $$$$=\:\mathrm{1}−\frac{\mathrm{1}}{\mathrm{40}}\:\:\:\:\:\Rightarrow\:\:\:\:\Omega\:=\:\frac{\mathrm{39}}{\mathrm{40}}…. \\ $$