I. Baleares 2023-A-E4

Demuestre que (para \(a \ge 0\)) $$\lim_{n \to\infty} \left [ \left ( 1+\frac{a}{n} \right ) \cdot \left ( 1+\frac{2a}{n} \right ) \dots \left ( 1+\frac{na}{n} \right ) \right ]^{\Large \frac{a}{n}} = \frac{(1+a)^{1+a}}{\E^a}.$$