Answer
$\lim\limits_{n \to \infty} a_n=0 $ and {$a_n$} is convergent
Work Step by Step
Consider $\lim\limits_{n \to \infty} a_n= \lim\limits_{n \to \infty} \dfrac{ n}{2^n}$
Check Indeterminate form of limit,we have $\lim\limits_{n \to \infty} \dfrac{ n}{2^n}=\dfrac{\infty}{\infty}$
We wiil have to use L-Hospital's rule.
That is, $\lim\limits_{n \to \infty} \dfrac{ n}{2^n}=\lim\limits_{n \to \infty} \dfrac{1}{(2^n \ln 2)}=0$
Thus, $\lim\limits_{n \to \infty} a_n=0 $ and {$a_n$} is convergent