u_(n+1) = u_n −u_n ^3 ; u_0 ∈ ]0, 1[ . show that u_n ∈ ]0, 1[ . show that u_n converges to 0 v_n = (1/u_(n+1) ) − (1/u_n ^2 ) . express v_n interms of u_n . show that v_n converges to 2 f(x) = ((2−x)/((1−x)^2 )) . show that f is increasing and deduce that v_n is decreasing . show that v


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Question Number 208418 by alcohol last updated on 15/Jun/24

$u_{n + 1} = u_{n} - u_{n}^{3}; u_{0} \in] 0, 1 [$ $. s h o w t h a t u_{n} \in] 0, 1 [$ $. s h o w t h a t u_{n} c o n v e r g e s t o 0$ $v_{n} = \frac{1}{u_{n + 1}} - \frac{1}{u_{n}^{2}}$ $. e x p r e s s v_{n} i n t e r m s o f u_{n}$ $. s h o w t h a t v_{n} c o n v e r g e s t o 2$ $f (x) = \frac{2 - x}{{(1 - x)}^{2}}$ $. s h o w t h a t f i s i n c r e a s i n g a n d d e d u c e t h a t$ $v_{n} i s d e c r e a s i n g$ $. s h o w t h a t v_{n} ⩾ 2$
	Answered by Berbere last updated on 15/Jun/24

	$t c h e k e x p r e s s i o n o f V_{n} = . . . ?$
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