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Question Number 152946 by john_santu last updated on 03/Sep/21

((4038)/(1+(1/3)+(1/6)+(1/(10))+(1/(15))+...+(1/(1+2+3+4+...+2019)))) =?

$$\:\:\:\frac{\mathrm{4038}}{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{3}}+\frac{\mathrm{1}}{\mathrm{6}}+\frac{\mathrm{1}}{\mathrm{10}}+\frac{\mathrm{1}}{\mathrm{15}}+...+\frac{\mathrm{1}}{\mathrm{1}+\mathrm{2}+\mathrm{3}+\mathrm{4}+...+\mathrm{2019}}}\:=? \\ $$

Commented by mathdanisur last updated on 03/Sep/21

▲ ((2∙2019)/(1 + (1/(1 + 2)) + (1/(1 + 2 + 3)) + ... + (1/(1 + 2 + 3 + ... + 2019)))) A = 1 + (1/(1 + 2)) + (1/(1 + 2 + 3)) + ... + (1/(1 + 2 + ... + 2019)) A = 1 + (2/(2∙3)) + (2/(3∙4)) + ... + (2/(2019∙2020)) (A/2) = (1/2) + (1/(2∙3)) + (1/(3∙4)) + ... + (1/(2019∙2020)) (A/2) = (1/2) + ((3 - 2)/(2∙3)) + ((4 - 3)/(3∙4)) + ... + ((2020 - 2019)/(2019∙2020)) (A/2) = (1/2) + (1/2) - (1/3) + (1/3) - (1/4) + ... + (1/(2019)) - (1/(2020)) (A/2) = 1 - (1/(2020)) ⇒ A = ((2∙2019)/(2020)) ⇒ (((2∙2019)/(2∙2019))/(2020)) = 2020

$$\blacktriangle\:\frac{\mathrm{2}\centerdot\mathrm{2019}}{\mathrm{1}\:+\:\frac{\mathrm{1}}{\mathrm{1}\:+\:\mathrm{2}}\:+\:\frac{\mathrm{1}}{\mathrm{1}\:+\:\mathrm{2}\:+\:\mathrm{3}}\:+\:...\:+\:\frac{\mathrm{1}}{\mathrm{1}\:+\:\mathrm{2}\:+\:\mathrm{3}\:+\:...\:+\:\mathrm{2019}}} \\ $$$$\boldsymbol{\mathrm{A}}\:=\:\mathrm{1}\:+\:\frac{\mathrm{1}}{\mathrm{1}\:+\:\mathrm{2}}\:+\:\frac{\mathrm{1}}{\mathrm{1}\:+\:\mathrm{2}\:+\:\mathrm{3}}\:+\:...\:+\:\frac{\mathrm{1}}{\mathrm{1}\:+\:\mathrm{2}\:+\:...\:+\:\mathrm{2019}} \\ $$$$\boldsymbol{\mathrm{A}}\:=\:\mathrm{1}\:+\:\frac{\mathrm{2}}{\mathrm{2}\centerdot\mathrm{3}}\:+\:\frac{\mathrm{2}}{\mathrm{3}\centerdot\mathrm{4}}\:+\:...\:+\:\frac{\mathrm{2}}{\mathrm{2019}\centerdot\mathrm{2020}} \\ $$$$\frac{\boldsymbol{\mathrm{A}}}{\mathrm{2}}\:=\:\frac{\mathrm{1}}{\mathrm{2}}\:+\:\frac{\mathrm{1}}{\mathrm{2}\centerdot\mathrm{3}}\:+\:\frac{\mathrm{1}}{\mathrm{3}\centerdot\mathrm{4}}\:+\:...\:+\:\frac{\mathrm{1}}{\mathrm{2019}\centerdot\mathrm{2020}} \\ $$$$\frac{\boldsymbol{\mathrm{A}}}{\mathrm{2}}\:=\:\frac{\mathrm{1}}{\mathrm{2}}\:+\:\frac{\mathrm{3}\:-\:\mathrm{2}}{\mathrm{2}\centerdot\mathrm{3}}\:+\:\frac{\mathrm{4}\:-\:\mathrm{3}}{\mathrm{3}\centerdot\mathrm{4}}\:+\:...\:+\:\frac{\mathrm{2020}\:-\:\mathrm{2019}}{\mathrm{2019}\centerdot\mathrm{2020}} \\ $$$$\frac{\boldsymbol{\mathrm{A}}}{\mathrm{2}}\:=\:\frac{\mathrm{1}}{\mathrm{2}}\:+\:\frac{\mathrm{1}}{\mathrm{2}}\:-\:\frac{\mathrm{1}}{\mathrm{3}}\:+\:\frac{\mathrm{1}}{\mathrm{3}}\:-\:\frac{\mathrm{1}}{\mathrm{4}}\:+\:...\:+\:\frac{\mathrm{1}}{\mathrm{2019}}\:-\:\frac{\mathrm{1}}{\mathrm{2020}} \\ $$$$\frac{\boldsymbol{\mathrm{A}}}{\mathrm{2}}\:=\:\mathrm{1}\:-\:\frac{\mathrm{1}}{\mathrm{2020}}\:\:\:\Rightarrow\:\:\boldsymbol{\mathrm{A}}\:=\:\frac{\mathrm{2}\centerdot\mathrm{2019}}{\mathrm{2020}} \\ $$$$\Rightarrow\:\frac{\frac{\mathrm{2}\centerdot\mathrm{2019}}{\mathrm{2}\centerdot\mathrm{2019}}}{\mathrm{2020}}\:=\:\mathrm{2020} \\ $$

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