My problem is actually deeper than just evaluation this limit, but learning how to solve the following limit will help me a lot:
$$\lim_{n\to \infty}\sqrt[n]{\frac{\ln(n)}{2^n+1}}$$
Just some context. I am studying Convergence Tests for Infinite Series. And as it is known, one of the tests is the root test. I understand how it works and I have a good feeling of when I can use it, but that implies that I must know how to solve this kind of limit. My textbook assumes that I already know how to solve this kind of limit and does not provide the step by step. I assume that most limits of this kind will have a similar strategy to solve. It would help me a lot if someone could help me. Specially if you could provide a solution without using l'Hopital.
Using Hagen's hint, for $n > e$ we have $$\frac{1}{\sqrt[n]{2}\cdot 2} = \sqrt[n]{\frac{1}{2\cdot 2^n} } < \sqrt[n]{\frac{\ln{n}}{2^n + 1}} < \sqrt[n]{\frac{n}{2^n}} = \frac{\sqrt[n]{n}}{2}.$$ You can now use the Squeeze Theorem, along with $\sqrt[n]{n} \to 1$ and $\sqrt[n]{2} \to 1$ as $n \to \infty$.