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-

Question Number 178561 by Huy last updated on 18/Oct/22

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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