\mathop {\lim }\limits_{n o \infty } \left[ {{1 \over {1 + n}} + {1 \over {2 + n}} + {1 \over {3 + n}}\, + \,...\, + \,{1 \over {2n}}} ight] is equal to

Question

\mathop {\lim }\limits_{n 	o \infty } \left[ {{1 \over {1 + n}} + {1 \over {2 + n}} + {1 \over {3 + n}}\, + \,...\, + \,{1 \over {2n}}} ight] is equal to

Accepted Answer

Answer: B. {\log _e}2 \begin{aligned} & \lim _{n ightarrow \infty}\left[\frac{1}{1+n}+\frac{1}{2+n}+\frac{1}{3+n}+\ldots \ldots+\frac{1}{2 n}ight] \\ & =\lim _{n ightarrow \infty} \sum_{r=1}^n \frac{1}{r+n} \\ & =\lim _{n ightarrow \infty} \sum_{r=1}^n \frac{1}{n}\left(\frac{1}{\frac{r}{n}+1}ight) \\ & =\int_0^1 \frac{d x}{x+1} \\ & =\left.\log _e(1+\mathrm{x})ight|_0 ^1 \\ & =\log _e^2 \end{aligned}

$$\mathop {\lim }\limits_{n \to \infty } \left[ {{1 \over {1 + n}} + {1 \over {2 + n}} + {1 \over {3 + n}}\, + \,...\, + \,{1 \over {2n}}} \right]$$ is equal to

Answer

Explanation

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