$$\mathop {\lim }\limits_{n \to \infty } \left[ {{1 \over {1 + n}} + {1 \over {2 + n}} + {1 \over {3 + n}}\, + \,...\, + \,{1 \over {2n}}} \right]$$ is equal to
Solution
$$
\begin{aligned}
& \lim _{n \rightarrow \infty}\left[\frac{1}{1+n}+\frac{1}{2+n}+\frac{1}{3+n}+\ldots \ldots+\frac{1}{2 n}\right] \\\\
& =\lim _{n \rightarrow \infty} \sum_{r=1}^n \frac{1}{r+n} \\\\
& =\lim _{n \rightarrow \infty} \sum_{r=1}^n \frac{1}{n}\left(\frac{1}{\frac{r}{n}+1}\right) \\\\
& =\int_0^1 \frac{d x}{x+1} \\\\
& =\left.\log _e(1+\mathrm{x})\right|_0 ^1 \\\\
& =\log _e^2
\end{aligned}
$$
About this question
Subject: Mathematics · Chapter: Definite Integration · Topic: Properties of Definite Integrals
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