Answer
Converges
Work Step by Step
Consider $a_n=\dfrac{n^{\sqrt 2}}{2^n}$
Now, $l=\lim\limits_{n \to \infty} |\dfrac{a_{n+1}}{a_{n}} |=\lim\limits_{n \to \infty}|\dfrac{\dfrac{(n+1)^{\sqrt 2}}{2^{n+1}}}{\dfrac{n^{\sqrt 2}}{2^n}}|$
Thus, we have $l=(\dfrac{1}{2})\lim\limits_{n \to \infty}(\dfrac{(n+1)}{n})^{\sqrt 2}=(\dfrac{1}{2})(1)$
So, $l=\dfrac{1}{2} \lt 1$
Hence, the series Converges absolutely by the ratio test.