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

Question Number 29461 by prof Abdo imad last updated on 09/Feb/18

l e t g i v e u_{n} = \frac{1}{\sqrt{n}} (\frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \dots + \frac{1}{\sqrt{n}})

f i n d {l i m}_{n \to + \infty} u_{n} .

Commented by prof Abdo imad last updated on 13/Feb/18

w e h a v e u_{n} = \frac{\sqrt{n}}{n} (\frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \dots . . + \frac{1}{\sqrt{n}})

= \frac{1}{n} (\frac{1}{\sqrt{\frac{1}{n}}} + \frac{1}{\sqrt{\frac{2}{n}}} + \dots . + \frac{1}{\sqrt{\frac{n}{n}}})

= \frac{1}{n} \sum_{k = 1}^{n} \frac{1}{\sqrt{\frac{k}{n}}} s o u_{n} i s a R i e m a n s u m a n d

{l i m}_{n \to \infty} u_{n} = \int_{0}^{1} \frac{d x}{\sqrt{x}} = {[2 \sqrt{x}]}_{0}^{1} = 2 .