Question

Simplify: $\underset{n\to \infty }{lim}\sum _{r=1}^n\frac{1}{n}{e}^{r/n}$

• A. e
• B. $e-1$
• C. $1-e$
• D. $e+1$

Answer 1

The correct option is B $e-1$

$\underset{n\to \infty }{lim}{\sum }_{r=1}^n\frac{1}{n}{e}^{r/n}$

Let $x=\frac{r}{n}⟹dx=\frac{1}{n}$ And when $r=1,x=0$ and when $r=n,x=1$

Hence,we get-

${\int }_0^1{e}^{x}dx={\left[e^x\right]}_0^1$

$=e-1$

Hence, answer is option-(B).