Let be $a,b> 0$ and $\gamma \in (0,1) $. Set $x_n = [\gamma \sqrt[n]{a} + (1-\gamma)\sqrt[n]{b} ]^n$ for each $n\in\mathbb{N}$.

My question is if this sequence $(x_n)$ is convergent and, if it so, which is its limit. I tried some ideas but none of them work. I guess I am stuck now, but it seems not a difficult sequence, so I'm sure somebody can give me a hint or a good idea here.

Thank you in advance.

Let be $a,b> 0$ and $\gamma \in (0,1) $. Set $x_n = (\gamma \sqrt[n]{a} + (1-\gamma)\sqrt[n]{b} )^n$ for each $n\in\mathbb{N}$.

My question is if this sequence $(x_n)$ is convergent and, if it so, which is its limit. I tried some ideas but none of them work. I guess I am stuck now, but it seems not a difficult sequence, so I'm sure somebody can give me a hint or a good idea here.

Thank you in advance.