2026 EJC P1 Q5

The diagram shows a sketch of the curve \(y=x^2e^x\) for \(x\geq0\), with \(n\) rectangles of equal width \(\frac1n\), where \(n\in\mathbb{Z}^+\), drawn under the curve for \(0\leq x\leq1\).

Let \(A_n\) denote the total area of the \(n\) rectangles.

1. Show that \(A_n=\frac{1}{n^3}\sum_{r=1}^{n-1}r^2e^{r/n}\).[3]
2. Find the exact value of \(\displaystyle\lim_{n\to\infty}A_n\).[4]