$\lim _{n \rightarrow \infty} \sum_{r=1}^{n} \frac{1}{n} e^{\frac{r}{n}}$ is

JEE Main · 2004 · 24 Apr 2004 · Offline · Definite Integrals

lim_{n \to \infty} \sum_{r = 1}^{n} \frac{1}{n} e^{\frac{r}{n}}

Answer: (B)

Read the sum

\sum_{r = 1}^{n} \frac{1}{n} e^{r / n}

as a Riemann sum for

e^{x}

[0, 1]

built from

n

equal subintervals. As

n \to \infty

it converges to

\int_{0}^{1} e^{x} d x = e^{1} - e^{0} = e - 1

JEE Main · 2004 · 24 Apr 2004 · Offline · Single correct option

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