let-u-n-1-n-k-1-n-1-n-4k-find-lim-n-u-n-

Question Number 40149 by maxmathsup by imad last updated on 16/Jul/18

l e t u_{n} = \frac{1}{\sqrt{n}} \sum_{k = 1}^{n} \frac{1}{\sqrt{n + 4 k}}

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

Commented by math khazana by abdo last updated on 17/Jul/18

w e h a v e u_{n} = \frac{1}{n} \sum_{k = 1}^{n} \frac{1}{\sqrt{1 + 4 \frac{k}{n}}}, 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{1 + 4 x}}

= {[\frac{1}{2} \sqrt{1 + 4 x}]}_{0}^{1} = \frac{1}{2} (\sqrt{5} - 1)