# How do you determine whether the sequence a_n=(n+1)^n/n^(n+1) converges, if so how do you find the limit?

Mar 29, 2017

${\lim}_{n \to \infty} {a}_{n} = 0$

#### Explanation:

Note that:

${\left(1 + \frac{1}{n}\right)}^{n}$

is monotonically increasing with limit $e$ as $n \to \infty$

So:

${\lim}_{n \to \infty} {a}_{n} = {\lim}_{n \to \infty} {\left(n + 1\right)}^{n} / \left({n}^{n + 1}\right)$

$\textcolor{w h i t e}{{\lim}_{n \to \infty} {a}_{n}} = {\lim}_{n \to \infty} {\left(\frac{n + 1}{n}\right)}^{n} / n$

$\textcolor{w h i t e}{{\lim}_{n \to \infty} {a}_{n}} = {\lim}_{n \to \infty} {\left(1 + \frac{1}{n}\right)}^{n} / n$

$\textcolor{w h i t e}{{\lim}_{n \to \infty} {a}_{n}} \le {\lim}_{n \to \infty} \frac{e}{n}$

$\textcolor{w h i t e}{{\lim}_{n \to \infty} {a}_{n}} = 0$