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Question Number 188482 by mathlove last updated on 02/Mar/23

512x^(1−x^(−3) ) =−1 find volue of Σ_(n=1) ^∞ (x^2 )^n =?

$$\mathrm{512}{x}^{\mathrm{1}−{x}^{−\mathrm{3}} } =−\mathrm{1} \\ $$$${find}\:\:{volue}\:\:{of}\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left({x}^{\mathrm{2}} \right)^{{n}} =? \\ $$

Answered by mr W last updated on 02/Mar/23

512x^(1−x^(−3) ) =−1 (512)^(−3) (x^(−3) )^(1−x^(−3) ) =(−1)^(−3) =−1 (x^(−3) )^(x^(−3) −1) =−512^(−3) =(−8)^(−9) ⇒x^(−3) =−8=(−2)^3 =(−(1/2))^(−3) ⇒x=−(1/2) ⇒x^2 =(1/4) Σ_(n=1) ^∞ x^(2n) =((1/4)/(1−(1/4)))=(1/3) ✓

$$\mathrm{512}{x}^{\mathrm{1}−{x}^{−\mathrm{3}} } =−\mathrm{1} \\ $$$$\left(\mathrm{512}\right)^{−\mathrm{3}} \left({x}^{−\mathrm{3}} \right)^{\mathrm{1}−{x}^{−\mathrm{3}} } =\left(−\mathrm{1}\right)^{−\mathrm{3}} =−\mathrm{1} \\ $$$$\left({x}^{−\mathrm{3}} \right)^{{x}^{−\mathrm{3}} −\mathrm{1}} =−\mathrm{512}^{−\mathrm{3}} =\left(−\mathrm{8}\right)^{−\mathrm{9}} \\ $$$$\Rightarrow{x}^{−\mathrm{3}} =−\mathrm{8}=\left(−\mathrm{2}\right)^{\mathrm{3}} =\left(−\frac{\mathrm{1}}{\mathrm{2}}\right)^{−\mathrm{3}} \\ $$$$\Rightarrow{x}=−\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\Rightarrow{x}^{\mathrm{2}} =\frac{\mathrm{1}}{\mathrm{4}} \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}{x}^{\mathrm{2}{n}} =\frac{\frac{\mathrm{1}}{\mathrm{4}}}{\mathrm{1}−\frac{\mathrm{1}}{\mathrm{4}}}=\frac{\mathrm{1}}{\mathrm{3}}\:\checkmark \\ $$

Commented by mathlove last updated on 02/Mar/23

i do′nt understand for this (512)^(−3) (x^(−3) )^(1−x^(−3) ) =(x^(−3) )^(x^(−3) −1) =−(512)^(−3) pleas discrib this

$${i}\:{do}'{nt}\:{understand}\:{for}\:{this} \\ $$$$\left(\mathrm{512}\right)^{−\mathrm{3}} \left({x}^{−\mathrm{3}} \right)^{\mathrm{1}−{x}^{−\mathrm{3}} } =\left({x}^{−\mathrm{3}} \right)^{{x}^{−\mathrm{3}} −\mathrm{1}} =−\left(\mathrm{512}\right)^{−\mathrm{3}} \\ $$$${pleas}\:{discrib}\:{this} \\ $$

Commented by mr W last updated on 02/Mar/23

(512)^(−3) (x^(−3) )^(1−x^(−3) ) =(−1)^(−3) =−1 clear? (x^(−3) )^(1−x^(−3) ) =−(512)^3 clear? (x^(−3) )^(−(1−x^(−3) )) =(−1)^(−3) (512)^(−3) clear? (x^(−3) )^(x^(−3) −1) =−(512)^(−3) clear? (x^(−3) )^(x^(−3) −1) =(−512)^(−3) clear? (x^(−3) )^(x^(−3) −1) =(−8^3 )^(−3) clear? (x^(−3) )^(x^(−3) −1) =(−8)^(−9) clear?

$$\left(\mathrm{512}\right)^{−\mathrm{3}} \left({x}^{−\mathrm{3}} \right)^{\mathrm{1}−{x}^{−\mathrm{3}} } =\left(−\mathrm{1}\right)^{−\mathrm{3}} =−\mathrm{1}\:\:\:{clear}? \\ $$$$\left({x}^{−\mathrm{3}} \right)^{\mathrm{1}−{x}^{−\mathrm{3}} } =−\left(\mathrm{512}\right)^{\mathrm{3}} \:\:\:{clear}? \\ $$$$\left({x}^{−\mathrm{3}} \right)^{−\left(\mathrm{1}−{x}^{−\mathrm{3}} \right)} =\left(−\mathrm{1}\right)^{−\mathrm{3}} \left(\mathrm{512}\right)^{−\mathrm{3}} \:\:\:{clear}? \\ $$$$\left({x}^{−\mathrm{3}} \right)^{{x}^{−\mathrm{3}} −\mathrm{1}} =−\left(\mathrm{512}\right)^{−\mathrm{3}} \:\:\:{clear}? \\ $$$$\left({x}^{−\mathrm{3}} \right)^{{x}^{−\mathrm{3}} −\mathrm{1}} =\left(−\mathrm{512}\right)^{−\mathrm{3}} \:\:\:{clear}? \\ $$$$\left({x}^{−\mathrm{3}} \right)^{{x}^{−\mathrm{3}} −\mathrm{1}} =\left(−\mathrm{8}^{\mathrm{3}} \right)^{−\mathrm{3}} \:\:\:{clear}? \\ $$$$\left({x}^{−\mathrm{3}} \right)^{{x}^{−\mathrm{3}} −\mathrm{1}} =\left(−\mathrm{8}\right)^{−\mathrm{9}} \:\:\:{clear}? \\ $$

Commented by mathlove last updated on 02/Mar/23

tanks mr very thank

$${tanks}\:{mr}\:{very}\:{thank} \\ $$

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