find-a-equivalent-of-U-n-1-1-2-1-3-1-n-

Question Number 91632 by mathmax by abdo last updated on 02/May/20

f i n d a e q u i v a l e n t o f U_{n} = 1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \dots + \frac{1}{\sqrt{n}}

Commented by mathmax by abdo last updated on 02/May/20

w e h s v e U_{n} = \sum_{k = 1}^{n} \frac{1}{\sqrt{k}} t h e s e q u e n c e n \to \frac{1}{\sqrt{n}} i s d e c r e a s i n g \Rightarrow

U_{n} \sim \int_{1}^{n} \frac{d t}{\sqrt{t}} = {[2 \sqrt{t}]}_{1}^{n} = 2 \sqrt{n} - 2 \Rightarrow U_{n} \sim 2 \sqrt{n} (n \to + \infty)