$\dpi{100} \lim_{n\rightarrow \infty }\sum_{r=1}^{n}\frac{1}{n}e^{r/n}\;\; \; \; is$ Option 1) $1-e$ Option 2) $e-1$ Option 3) $e$ Option 4) $e+1$

As we learnt in

Walli's Method -

Definite integral by first principle

$\int_{a}^{b}f(x)dx= \left ( b-a \right )\lim_{n \to \infty }\frac{1}{n}\left [ f(a) +f(a+h)+f(a+2h)....\right ]$

where

$h=\frac{b-a}{n}$

- wherein

$\lim_{n\rightarrow \infty} \sum_{r=1}^{n}\frac{1}{n}e^{r/n}$

$\Rightarrow \int_{0}^{1}e^n dn$

$\Rightarrow \left [ e^n \right ]_{0}^{1}$

$\Rightarrow e-1$

Option 1)

$1-e$

This is incorrect option

Option 2)

$e-1$

This is correct option

Option 3)

$e$

This is incorrect option

Option 4)

$e+1$

This is incorrect option

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