Question Number 126777 by snipers237 last updated on 24/Dec/20
![let consider (u_n ) such as u_0 ∈]0;1[ and u_(n+1) =u_n −u_n ^2 1)Prove that lim_(n→∞) ^n (√u_n ) = 1 and that the convergence domain of Σu_n x^n is D=[−1;1[ 2) Prove that the one of Σu_n ^2 x^n is I=[−1;1]](Q126777.png)
$$\left.{let}\:{consider}\:\left({u}_{{n}} \right)\:{such}\:{as}\:{u}_{\mathrm{0}} \in\right]\mathrm{0};\mathrm{1}\left[\:{and}\:{u}_{{n}+\mathrm{1}} ={u}_{{n}} −{u}_{{n}} ^{\mathrm{2}} \:\right. \\ $$$$\left.\mathrm{1}\right){Prove}\:{that}\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:^{{n}} \sqrt{{u}_{{n}} }\:=\:\mathrm{1}\:{and}\:{that}\:{the}\:{convergence}\:{domain}\:{of}\:\Sigma{u}_{{n}} {x}^{{n}} \: \\ $$$${is}\:\:{D}=\left[−\mathrm{1};\mathrm{1}\left[\:\right.\right. \\ $$$$\left.\mathrm{2}\right)\:{Prove}\:{that}\:{the}\:{one}\:{of}\:\Sigma{u}_{{n}} ^{\mathrm{2}} {x}^{{n}} \:{is}\:\:{I}=\left[−\mathrm{1};\mathrm{1}\right] \\ $$