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Question

Question: \(\lim_{n \rightarrow \infty}\sum_{r = 1}^{n}{\frac{1}{n}e^{\frac{r}{n}}}\)is...

lim⁡n→∞∑r=1n1nern\lim_{n \rightarrow \infty}\sum_{r = 1}^{n}{\frac{1}{n}e^{\frac{r}{n}}}is

A

e+1e + 1

B

e−1e - 1

C

1−e1 - e

D

ee

Answer

e−1e - 1

Explanation

Solution

lim⁡n→∞∑r=1n1nern=∫01exdx=[ex]01=e−1\lim_{n \rightarrow \infty}\sum_{r = 1}^{n}{\frac{1}{n}e^{\frac{r}{n}} = \int_{0}^{1}{e^{x}dx = \lbrack e^{x}\rbrack_{0}^{1} = e - 1}}.