# Convergence of the limit of the $n$th root of a term

In my course I often see questions that ask me to calculate the limit of sequences such as: $$\lim\limits_{n \to \infty}{\sqrt [n]{a_n}}$$

How do I handle these questions?

A related question is to show that as ${a_n\to\infty}$ then $${\sqrt [n]{a_n}} > \left(1+\frac {1}{n}\right)$$

for almost every $n$.

I don't know the answer to the first question, so I'm having trouble with the second.

Thanks

• For the second part, consider $\left(1+\dfrac1n\right)^n \lt e$ for positive integer $n$ but $e$ is the limit as $n\to \infty$ – Henry Apr 20 '15 at 20:06
• For the first question, it heavily depends on the actual sequence $|a_n|$. Nontheless, a useful trick to exploit is studying whether $L:=\lim_{n\rightarrow+\infty}\frac{\ln|a_n|}{n}$ exists and consider $e^L$. For instance, this trick lets you show that $\lim_{n\rightarrow+\infty}\sqrt[n]{|p(n)|}=1$ if $p$ is a polynomial. – user228113 Apr 20 '15 at 20:12

Quite a number of sequences $(a_n)$ will be analogous to the one in question 1243909, and any way to compute the limit will be similar.
If you can show that your sequence $a_n$ is bounded below by $x^{f(n)}$ and above by $k x^{f(n)}$, then the limit of $a_n ^ {(1/n)}$ is the limit of $x^{f(n)/n}$ if it exists.