Question Number 163849 by alephzero last updated on 11/Jan/22
$$\frac{\mathrm{1}}{\mathrm{1}\:\centerdot\:\mathrm{2}}\:+\:\frac{\mathrm{1}}{\mathrm{2}\:\centerdot\:\mathrm{3}}\:+\:…\:\frac{\mathrm{1}}{\mathrm{19}\:\centerdot\:\mathrm{20}}\:+\:\frac{\mathrm{1}}{\mathrm{20}\:\centerdot\:\mathrm{21}}\:=\:? \\ $$
Answered by cortano1 last updated on 11/Jan/22
$$\:\underset{{k}=\mathrm{1}} {\overset{\mathrm{20}} {\sum}}\:\frac{\mathrm{1}}{{k}\left({k}+\mathrm{1}\right)}\:=\:\underset{{k}=\mathrm{1}} {\overset{\mathrm{20}} {\sum}}\:\left(\frac{\mathrm{1}}{{k}}−\frac{\mathrm{1}}{{k}+\mathrm{1}}\right) \\ $$$$\:=\:\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}\right)+\left(\frac{\mathrm{1}}{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{3}}\right)+…+\left(\frac{\mathrm{1}}{\mathrm{20}}−\frac{\mathrm{1}}{\mathrm{21}}\right) \\ $$$$\:=\:\mathrm{1}−\frac{\mathrm{1}}{\mathrm{21}}\:=\:\frac{\mathrm{20}}{\mathrm{21}} \\ $$
Commented by alephzero last updated on 11/Jan/22
$$\mathrm{Thank}\:\mathrm{You}\:\mathrm{very}\:\mathrm{much},\:\mathrm{sir}! \\ $$