# How do you determine if a_n=(1+n)^(1/n) converge and find the limits when they exist?

Nov 8, 2017

The series converge to $= 1$

#### Explanation:

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

Taking logarithms

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

Therefore,

$y = {e}^{\frac{1}{n} \ln \left(1 + n\right)}$

So,

${\lim}_{n \to \infty} {u}_{n} = {\lim}_{n \to \infty} {e}^{\frac{1}{n} \ln \left(1 + n\right)}$

Apply the limit chain rule

${\lim}_{n \to \infty} \left(\frac{1}{n} \ln \left(1 + n\right)\right) = {\lim}_{n \to \infty} \left(\frac{1}{1 + n}\right)$ ( by L'Hôpital's rule)

$= 0$

Therefore,

${\lim}_{n \to \infty} {e}^{\frac{1}{n} \ln \left(1 + n\right)} = {\lim}_{n \to \infty} {e}^{0} = 1$