let give u_n =(√(ln(n+1)−ln(n))) 1)give a simple eqivalent of u_n (n→∞) 2) deduce the nature of u


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Question Number 31522 by abdo imad last updated on 09/Mar/18

$l e t g i v e u_{n} = \sqrt{l n (n + 1) - l n (n)}$ $1) g i v e a s i m p l e e q i v a l e n t o f u_{n} (n \to \infty)$ $2) d e d u c e t h e n a t u r e o f u_{n} .$
Commented by abdo imad last updated on 15/Mar/18

$w e h a v e u_{n} = {(l n (\frac{n + 1}{n}))}^{\frac{1}{2}} = {(l n (1 + \frac{1}{n}))}^{\frac{1}{2}} b u t$ $l n (1 + \frac{1}{n}) \sim \frac{1}{n} (n \to \infty) \Rightarrow u_{n} \sim \frac{1}{\sqrt{n}} .$ $2) f r o m Q . 1) (u_{n}) i s c o n v e r g e n t a n d {l i m}_{n \to \infty} u_{n} = 0 .$
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