202111221615 Exercises 9.1.A (Q1)

For Exercises 1-8, determine if the given sequence is convergent. If so then find its limit.

$\{ ne^{-n}\}_{n=0}^{\infty}$

Solution.

By L’Hôpital’s Rule, treating an integer $n \ge 0$ as a real-valued variable $x$ ,

$\begin{aligned} & \quad \lim_{n\to\infty} \frac{n}{e^n} \\ & = \lim_{n\to\infty} \frac{0}{e^n} \\ & = 0 \\ \end{aligned}$

Thus the sequence is convergent and its limit is $0$ .