Question:
Evaluate $\lim_{n\to\infty} \frac{1}{n^2}\sum_{r=1}^n r e^{r/n}$
Evaluate $\lim_{n\to\infty} \frac{1}{n^2}\sum_{r=1}^n r e^{r/n}$
Show Hint
Convert summations into Riemann integrals using $\frac{r}{n} \to x$.
Updated On: Jun 10, 2026
- $0$
- $1$
- $e$
- $2e$
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The Correct Option is B
Solution and Explanation
\[
L = \frac{1}{n}\sum \left(\frac{r}{n}\right)e^{r/n}
\]
Let \( x = \frac{r}{n} \)
\[
L = \int_0^1 x e^x dx
\]
Integration by parts:
\[
= e^x(x-1)\Big|_0^1
= 0 - (-1) = 1
\]
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