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-u-n-x-n-is-D-1-1-2-Prove-that-the-one-of-u-n-2-x-n

Question Number 126777 by snipers237 last updated on 24/Dec/20

l e t c o n s i d e r (u_{n}) s u c h a s u_{0} \in] 0; 1 [a n d u_{n + 1} = u_{n} - u_{n}^{2}

1) P r o v e t h a t \lim_{n \to \infty}^{n} \sqrt{u_{n}} = 1 a n d t h a t t h e c o n v e r g e n c e d o m a i n o f Σ u_{n} x^{n}

i s D = [- 1; 1 [

2) P r o v e t h a t t h e o n e o f Σ u_{n}^{2} x^{n} i s I = [- 1; 1]