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  4. The value of displaystyle lim n arrow ∞((1/n)+(e1 / n/n)+(e2 / n/n)+ ldots+(e(n-1) / n/n)) is

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

Limits and Derivatives

Solution:

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