Question Number 36576 by ast empire last updated on 03/Jun/18
Commented by prof Abdo imad last updated on 03/Jun/18
$${let}\:{put}\:{A}\:=\:\frac{{x}^{\mathrm{100}} }{\mathrm{1}+{x}+{x}^{\mathrm{2}} \:+….+{x}^{\mathrm{200}} } \\ $$$${if}\:{x}=\mathrm{1}\:\:\:{A}\:=\:\frac{\mathrm{1}}{\mathrm{201}} \\ $$$${if}\:{x}\:\neq\mathrm{1}\:\:{A}\:=\:\frac{{x}^{\mathrm{100}} }{\frac{\mathrm{1}−{x}^{\mathrm{201}} }{\mathrm{1}−{x}}}\:=\frac{{x}^{\mathrm{100}} \left(\mathrm{1}−{x}\right)}{\mathrm{1}−{x}^{\mathrm{201}} }\:\Rightarrow \\ $$$${A}\:=\:\frac{{x}^{\mathrm{100}} \:−{x}^{\mathrm{101}} }{\mathrm{1}−{x}^{\mathrm{201}} }\:. \\ $$
Answered by ajfour last updated on 03/Jun/18
$$=\frac{{x}^{\mathrm{100}} \left(\mathrm{1}−{x}\right)}{\mathrm{1}−{x}^{\mathrm{201}} }\:\:\:\:\:\left({provided}\:\mid{x}\mid<\mathrm{1}\:\right)\:. \\ $$