find-the-nature-of-u-n-u-n-1-2-n-n-3-

Question Number 32041 by abdo imad last updated on 18/Mar/18

f i n d t h e n a t u r e o f Σ u_{n} /

u_{n} = \frac{\sqrt{1} + \sqrt{2} + \dots . + \sqrt{n}}{n^{3}} .

Commented by abdo imad last updated on 20/Mar/18

w e h a v e u_{n} = \frac{1}{n^{2}} (\frac{1}{\sqrt{n}} \sum_{k = 1}^{n} \sqrt{\frac{k}{n}})

= \frac{1}{n \sqrt{n}} (\frac{1}{n} \sum_{k = 1}^{n} \sqrt{\frac{k}{n}}) b u t {l i m}_{n \to \infty} \frac{1}{n} \sum_{k = 1}^{n} \sqrt{\frac{k}{n}}

= \int_{0}^{1} \sqrt{x} d x = {[\frac{2}{3} x^{\frac{3}{2}}]}_{0}^{1} = \frac{2}{3} \Rightarrow u_{n} \sim \frac{2}{3 n \sqrt{n}} a n d

t h e s e r i e \sum_{n ⩾ 1} \frac{2}{3 n \sqrt{n}} i s c o n v e r g e n t s o Σ u_{n} c o n v e r g e s .