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  4. displaystyle lim n arrow ∞ (1/n)(1+e1 / n+e2 / n+ ldots+e(n-1/n)) is equal to

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Q. $\displaystyle\lim _{n \rightarrow \infty} \frac{1}{n}\left(1+e^{1 / n}+e^{2 / n}+\ldots+e^{\frac{n-1}{n}}\right)$ is equal to

Limits and Derivatives

Solution:

$\displaystyle\lim _{n \rightarrow \infty} \frac{1}{n}\left(1+e^{1 / n}+e^{2 / n}+\ldots+e^{\frac{n-1}{n}}\right)$
$=\displaystyle\lim _{n \rightarrow \infty} \frac{1}{n} \cdot \frac{1-\left(e^{1 / n}\right)^{n}}{1-e^{1 / n}}$
$=\displaystyle\lim _{n \rightarrow \infty} \frac{1-e}{n\left(1-1-\frac{1}{n}-\frac{1}{2 !} \cdot \frac{1}{n^{2}} \ldots\right)}$
$=\displaystyle\lim _{n \rightarrow \infty} \frac{1-e}{-1-\frac{1}{2 !} \cdot \frac{1}{n} \ldots}$
$=\frac{1-e}{-1}=e-1$