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

Feb 8, 2017

${\lim}_{n \to \infty} \sqrt[n]{n} = 1$

#### Explanation:

As the terms of the sequence:

${a}_{n} = \sqrt[n]{n}$

are positive, we can consider the sequence:

${b}_{n} = \ln {a}_{n} = \ln \sqrt[n]{n}$

Using the properties of logarithms we have:

${b}_{n} = \ln \sqrt[n]{n} = \frac{\ln n}{n}$

Now consider the function:

$f \left(x\right) = \ln \frac{x}{x}$

if ${\lim}_{x \to \infty} f \left(x\right)$ exists, it must be the same as ${\lim}_{n \to \infty} {b}_{n}$, since ${b}_{n} = f \left(n\right)$.

The limit ${\lim}_{x \to \infty} \ln \frac{x}{x}$ is in the indeterminate form $\frac{\infty}{\infty}$ and we can determine it using l'Hospital's rule:

${\lim}_{x \to \infty} \frac{\ln}{x} = {\lim}_{x \to \infty} \frac{\frac{d}{\mathrm{dx}} \ln x}{\frac{d}{\mathrm{dx}} x} = {\lim}_{x \to \infty} \frac{1}{x} = 0$

and we have established that:

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

As $\ln x$ is a continuous function we can also state that:

${\lim}_{n \to \infty} \ln {a}_{n} = \ln \left({\lim}_{n \to \infty} {a}_{n}\right) = 0$

which implies:

${\lim}_{n \to \infty} {a}_{n} = 1$