Answer
Converges
Work Step by Step
In order to solve the given series we will take the help of Ratio Test. This test states that when the limit $L \lt 1$, the series converges and for $L \gt 1$, the series diverges.
Let us consider $a_n=\dfrac{n^{\sqrt 2}}{2^n}$
Then $L=\lim\limits_{n \to \infty} |\dfrac{a_{n+1}}{a_{n}} |=\lim\limits_{n \to \infty}|\dfrac{\dfrac{(n+1)^{\sqrt 2}}{2^{n+1}}}{\dfrac{n^{\sqrt 2}}{2^n}}|$
$\implies \lim\limits_{n \to \infty}(\dfrac{1}{2})(\dfrac{n+1}{n})^{\sqrt 2}=\dfrac{1}{2} \lt 1$
Hence, the series Converges by the ratio test.