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Question Number 36576 by ast empire last updated on 03/Jun/18

Commented by prof Abdo imad last updated on 03/Jun/18

let put A = (x^(100) /(1+x+x^2 +....+x^(200) )) if x=1 A = (1/(201)) if x ≠1 A = (x^(100) /((1−x^(201) )/(1−x))) =((x^(100) (1−x))/(1−x^(201) )) ⇒ A = ((x^(100) −x^(101) )/(1−x^(201) )) .

$${let}\:{put}\:{A}\:=\:\frac{{x}^{\mathrm{100}} }{\mathrm{1}+{x}+{x}^{\mathrm{2}} \:+....+{x}^{\mathrm{200}} } \\ $$$${if}\:{x}=\mathrm{1}\:\:\:{A}\:=\:\frac{\mathrm{1}}{\mathrm{201}} \\ $$$${if}\:{x}\:\neq\mathrm{1}\:\:{A}\:=\:\frac{{x}^{\mathrm{100}} }{\frac{\mathrm{1}−{x}^{\mathrm{201}} }{\mathrm{1}−{x}}}\:=\frac{{x}^{\mathrm{100}} \left(\mathrm{1}−{x}\right)}{\mathrm{1}−{x}^{\mathrm{201}} }\:\Rightarrow \\ $$$${A}\:=\:\frac{{x}^{\mathrm{100}} \:−{x}^{\mathrm{101}} }{\mathrm{1}−{x}^{\mathrm{201}} }\:. \\ $$

Answered by ajfour last updated on 03/Jun/18

=((x^(100) (1−x))/(1−x^(201) )) (provided ∣x∣<1 ) .

$$=\frac{{x}^{\mathrm{100}} \left(\mathrm{1}−{x}\right)}{\mathrm{1}−{x}^{\mathrm{201}} }\:\:\:\:\:\left({provided}\:\mid{x}\mid<\mathrm{1}\:\right)\:. \\ $$

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