limn→∞ 1n1+e1/n+e2/n+e3/n+….+en−1n is equal to

A
$e$
B
$- e$
C
$e - 1$
D
$1 - e$

Solution:

We have,

$lim_{n \to \infty} \frac{1}{n} \{1 + e^{1 / n} + e^{2 / n} + e^{3 / n} + \dots + e^{\frac{n - 1}{n}}\}$

$= lim_{n \to \infty} \frac{1}{n} \{\frac{1 - {(e^{1 / n})}^{n}}{1 - e^{1 / n}}\} = lim_{n \to \infty} \frac{1 - e}{(\frac{1 - e^{1 / n}}{1 / n})} = \frac{1 - e}{- 1} = e - 1 .$

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