A sequence is given by { ((u_1 =1)),((u_(n+1) =(√(u_n ^2 +1))−u_n )) :} (n∈N, n≥1) Find lim u


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Question Number 178561 by Huy last updated on 18/Oct/22
$A sequence is given by { ((u_1 =1)),((u_(n+1) =(√(u_n ^2 +1))−u_n )) :} (n∈N, n≥1) Find lim u_n ?$
$A s e q u e n c e i s g i v e n b y$ ${\begin{cases} u_{1} = 1 \\ u_{n + 1} = \sqrt{u_{n}^{2} + 1} - u_{n} \end{cases} (n \in N, n ⩾ 1)$ $F i n d l i m u_{n} ?$
	Answered by mr W last updated on 18/Oct/22

	$\frac{u_{n + 1}}{u_{n}} = \sqrt{1 + {(\frac{1}{u_{n}})}^{2}} - 1 > 0$ $L = \sqrt{L^{2} + 1} - L$ $3 L^{2} = 1$ $\Rightarrow L = \frac{1}{\sqrt{3}} = \lim_{n \to \infty} u_{n}$
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