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Question Number 142646 by naka3546 last updated on 03/Jun/21

Prove that (1+(1/2^3 ))(1+(1/3^3 ))(1+(1/4^3 )) … < 3

$${Prove}\:\:{that} \\ $$ $$\:\:\:\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{3}} }\right)\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{3}} }\right)\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{4}^{\mathrm{3}} }\right)\:\ldots\:<\:\mathrm{3} \\ $$

Answered by 1549442205PVT last updated on 04/Jun/21

l.h.s=((3(2^2 −2+1).4(3^2 −3+1).5(4^2 −4+1)....)/((2.3.4...)^3 )) <((3.2^2 .4.3^2 5.4^2 .....)/(2^3 .(3.4....)^3 ))=((2^2 .5.(3.4....)^3 )/(2^3 (3.4...)^3 ))=(5/2)<3

$$\mathrm{l}.\mathrm{h}.\mathrm{s}=\frac{\mathrm{3}\left(\mathrm{2}^{\mathrm{2}} −\mathrm{2}+\mathrm{1}\right).\mathrm{4}\left(\mathrm{3}^{\mathrm{2}} −\mathrm{3}+\mathrm{1}\right).\mathrm{5}\left(\mathrm{4}^{\mathrm{2}} −\mathrm{4}+\mathrm{1}\right)....}{\left(\mathrm{2}.\mathrm{3}.\mathrm{4}...\right)^{\mathrm{3}} } \\ $$ $$<\frac{\mathrm{3}.\mathrm{2}^{\mathrm{2}} .\mathrm{4}.\mathrm{3}^{\mathrm{2}} \mathrm{5}.\mathrm{4}^{\mathrm{2}} .....}{\mathrm{2}^{\mathrm{3}} .\left(\mathrm{3}.\mathrm{4}....\right)^{\mathrm{3}} }=\frac{\mathrm{2}^{\mathrm{2}} .\mathrm{5}.\left(\mathrm{3}.\mathrm{4}....\right)^{\mathrm{3}} }{\mathrm{2}^{\mathrm{3}} \left(\mathrm{3}.\mathrm{4}...\right)^{\mathrm{3}} }=\frac{\mathrm{5}}{\mathrm{2}}<\mathrm{3} \\ $$

Commented bynaka3546 last updated on 04/Jun/21

Dank you ,,, thank you

$$\mathrm{Dank}\:\mathrm{you}\:,,,\:\mathrm{thank}\:\mathrm{you} \\ $$

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