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Question Number 161005 by vvvv last updated on 10/Dec/21

A=(1/(1×2))+(1/(3×4))+(1/(5×6))+....+(1/(37×38))+(1/(39×40)) B=(1/(21×40))+(1/(22×39))+(1/(23×38))+....+(1/(39×22))+(1/(40×21)) (A/B)=?

$$\boldsymbol{{A}}=\frac{\mathrm{1}}{\mathrm{1}×\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{3}×\mathrm{4}}+\frac{\mathrm{1}}{\mathrm{5}×\mathrm{6}}+....+\frac{\mathrm{1}}{\mathrm{37}×\mathrm{38}}+\frac{\mathrm{1}}{\mathrm{39}×\mathrm{40}} \\ $$$$\boldsymbol{{B}}=\frac{\mathrm{1}}{\mathrm{21}×\mathrm{40}}+\frac{\mathrm{1}}{\mathrm{22}×\mathrm{39}}+\frac{\mathrm{1}}{\mathrm{23}×\mathrm{38}}+....+\frac{\mathrm{1}}{\mathrm{39}×\mathrm{22}}+\frac{\mathrm{1}}{\mathrm{40}×\mathrm{21}} \\ $$$$\frac{\boldsymbol{{A}}}{\boldsymbol{{B}}}=? \\ $$

Answered by Raxreedoroid last updated on 11/Dec/21

A=Σ_(k=1) ^(39) (1/(k(k+1)))=Σ_(k=1) ^(39) (1/k)−(1/(k+1))=1−(1/(40)) B=Σ_(k=21) ^(40) (1/(k(61−k)))=(1/(61))Σ_(k=21) ^(40) (1/k)−(1/(k−61))=(1/(21))+(1/(40))+...(1/(40))+(1/(21)) =2Σ_(k=21) ^(30) (1/k)−(1/(k−61))=2Σ_(k=21) ^(40) (1/k)≈1.361606 (A/B)=(((39)/(40))/(1.361606))≈0.716065

$${A}=\underset{{k}=\mathrm{1}} {\overset{\mathrm{39}} {\sum}}\frac{\mathrm{1}}{{k}\left({k}+\mathrm{1}\right)}=\underset{{k}=\mathrm{1}} {\overset{\mathrm{39}} {\sum}}\frac{\mathrm{1}}{{k}}−\frac{\mathrm{1}}{{k}+\mathrm{1}}=\mathrm{1}−\frac{\mathrm{1}}{\mathrm{40}} \\ $$$${B}=\underset{{k}=\mathrm{21}} {\overset{\mathrm{40}} {\sum}}\frac{\mathrm{1}}{{k}\left(\mathrm{61}−{k}\right)}=\frac{\mathrm{1}}{\mathrm{61}}\underset{{k}=\mathrm{21}} {\overset{\mathrm{40}} {\sum}}\frac{\mathrm{1}}{{k}}−\frac{\mathrm{1}}{{k}−\mathrm{61}}=\frac{\mathrm{1}}{\mathrm{21}}+\frac{\mathrm{1}}{\mathrm{40}}+...\frac{\mathrm{1}}{\mathrm{40}}+\frac{\mathrm{1}}{\mathrm{21}} \\ $$$$=\mathrm{2}\underset{{k}=\mathrm{21}} {\overset{\mathrm{30}} {\sum}}\frac{\mathrm{1}}{{k}}−\frac{\mathrm{1}}{{k}−\mathrm{61}}=\mathrm{2}\underset{{k}=\mathrm{21}} {\overset{\mathrm{40}} {\sum}}\frac{\mathrm{1}}{{k}}\approx\mathrm{1}.\mathrm{361606} \\ $$$$\frac{{A}}{{B}}=\frac{\frac{\mathrm{39}}{\mathrm{40}}}{\mathrm{1}.\mathrm{361606}}\approx\mathrm{0}.\mathrm{716065} \\ $$

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