U-n-k-0-n-1-2k-1-find-a-eqivalent-of-U-n-n-

Question Number 144218 by Mathspace last updated on 23/Jun/21

U_{n} = \sum_{k = 0}^{n} \frac{1}{\sqrt{2 k + 1}}

f i n d a e q i v a l e n t o f U_{n} (n \to \infty)

Answered by ArielVyny last updated on 24/Jun/21

0 ⩽ k ⩽ n \to 0 ⩽ 2 k ⩽ 2 n

\to 1 ⩽ 2 k + 1 ⩽ 2 n + 1

\to \frac{1}{\sqrt{2 n + 1}} ⩽ \frac{1}{\sqrt{2 k + 1}} ⩽ 1

\to \frac{n + 1}{\sqrt{2 n + 1}} ⩽ U_{n} ⩽ n + 1