Question

Evaluate the following
$\large\underline{\sf{evaluate \: the \: following}}$
$lim_{n} = \infty ( \frac{1}{1 + \sqrt{1} } + \frac{1}{2 + \sqrt{2n} } + \frac{1}{3 + \sqrt{3n} } + - - - + \frac{1}{n + \sqrt{ {n}^{2} } }$
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Answer 1

Step-by-step explanation:

$\\ \tiny \tt \lim \limits_{n \to\infty} \left( \frac{1}{1 + \sqrt{n} } + \frac{1}{2 + \sqrt{2n} } + \frac{1}{3 + \sqrt{3n} } + - - - + \frac{1}{n + \sqrt{ {n}^{2} } } \right)$

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Required limit

$\\ \[ \begin{array}{l} \displaystyle \tt =\lim _{n \rightarrow \infty} \sum_{r=1}^{n} \frac{1}{r+\sqrt{r n}} \\ \\ \displaystyle \tt=\lim _{n \rightarrow \infty} \sum_{r=1}^{n} \frac{\frac{1}{n}}{\frac{r}{n}+\sqrt{\frac{r}{n}}} \\ \\ \displaystyle \tt=\int_{0}^{1} \frac{d x}{x+\sqrt{x}}=2 \int_{0}^{i} \frac{d x}{2 \sqrt{x}(\sqrt{x}+1)} \\ \\ \displaystyle \tt=2[\log (\sqrt{x}+1)]_{0}^{1} \\ \\ \displaystyle \tt=2 \log 2=\log _{e} 4 \end{array} \]$