let-u-n-k-1-n-1-k-find-a-equivalent-of-u-n-n-

Question Number 133121 by abdomsup last updated on 19/Feb/21

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

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

Answered by mindispower last updated on 19/Feb/21

\underset{k = 1}{\sum^{n}} \frac{1}{\sqrt{k + 1}} ⩽ \underset{k = 1}{\sum^{n}} \int_{k}^{k + 1} \frac{d x}{\sqrt{x}} ⩽ \underset{k = 1}{\sum^{n}} \frac{1}{\sqrt{k}}

\Rightarrow S_{n} - 1 ⩽ \int_{1}^{n} \frac{d x}{\sqrt{x}}, \int_{1}^{n + 1} \frac{d x}{\sqrt{x}} ⩽ S_{n}

S_{n} ⩽ 1 + 2 \sqrt{n} - 2

s_{n} ⩽ 2 \sqrt{n} - 1, 2 \sqrt{n + 1} - 1 ⩽ s_{n}

\frac{\sqrt{n + 1}}{\sqrt{n}} - \frac{1}{2 \sqrt{n}} ⩽ \frac{s_{n}}{2 \sqrt{n}} ⩽ 1 - \frac{1}{2 \sqrt{n}}

n \to \infty

\Rightarrow S_{n} \sim 2 \sqrt{n}

Commented by mathmax by abdo last updated on 19/Feb/21

thanks sir mind