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Question Number 30193 by abdo imad last updated on 17/Feb/18
![let p_n (x)=−1 +Σ_(k=1) ^k x^k 1) prove that the equation p_n (x)=0 have only one solution x_n ∈[0,1] . 2) prove that (x_n ) is decreasing and minored by (1/2) 3) prove that lim_(n→∞) x_n =(1/2) .](Q30193.png)
$${let}\:{p}_{{n}} \left({x}\right)=−\mathrm{1}\:+\sum_{{k}=\mathrm{1}} ^{{k}} \:{x}^{{k}} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{the}\:{equation}\:{p}_{{n}} \left({x}\right)=\mathrm{0}\:{have}\:{only}\:{one}\: \\ $$$${solution}\:{x}_{{n}} \in\left[\mathrm{0},\mathrm{1}\right]\:. \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\left({x}_{{n}} \right)\:{is}\:{decreasing}\:{and}\:{minored}\:{by}\:\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\left.\mathrm{3}\right)\:{prove}\:{that}\:{lim}_{{n}\rightarrow\infty} \:{x}_{{n}} =\frac{\mathrm{1}}{\mathrm{2}}\:. \\ $$