let-u-n-e-1-1-n-n-n-2-2-n-2-1-find-lim-u-n-

Question Number 32364 by prof Abdo imad last updated on 23/Mar/18

l e t u_{n} = {(e - {(1 + \frac{1}{n})}^{n})}^{\sqrt{n^{2} + 2} - \sqrt{n^{2} + 1}}

f i n d l i m u_{n}

Commented by prof Abdo imad last updated on 06/Apr/18

w e h a v e A_{n} = \sqrt{n^{2} + 2} - \sqrt{n^{2} + 1}

= n (\sqrt{1 + \frac{2}{n}} - \sqrt{1 + \frac{1}{n}}) b u t

\sqrt{1 + \frac{2}{n}} \sim 1 + \frac{1}{n} a n d \sqrt{1 + \frac{1}{n}} \sim 1 + \frac{1}{2 n} \Rightarrow

A_{n} \sim n + 1 - n - \frac{1}{2} \Rightarrow A_{n} \sim \frac{1}{2} a l s o w e h a v e

{(1 + \frac{1}{n})}^{n} = e^{n l n (1 + \frac{1}{n})}

{l n (1 + x))}^{^{'}} = \frac{1}{1 + x} = 1 - x + o (x^{2}) f o r x \in V (o)

l n (1 + x) = x - \frac{x^{2}}{2} + o (x^{3}) \Rightarrow

l n (1 + \frac{1}{n}) = \frac{1}{n} - \frac{1}{2 n^{2}} + o (\frac{1}{n^{3}}) \Rightarrow

n l n (1 + \frac{1}{n}) = 1 - \frac{1}{2 n} + o (\frac{1}{n^{2}}) \Rightarrow

e^{n l n (1 + \frac{1}{n})} = e^{1 - \frac{1}{2 n} + o (\frac{1}{n^{2}})} = e . (1 - \frac{1}{2 n} + o (\frac{1}{n^{2}})) \Rightarrow

e - {(1 + \frac{1}{n})}^{n} \sim \frac{e}{2 n} + o (\frac{1}{n^{2}}) \Rightarrow u_{n} \sim \sqrt{\frac{e}{2 n}} \Rightarrow

{l i m}_{n \to \infty} u_{n} = 0 .