About the sequence $n^{\frac{1}{n}} -1$

To tackle the fourth question, you can let

$$f(x)=x^\frac{1}{x} - \frac{1}{\sqrt{x}}.$$

For $x^\frac{1}{x}$, it is easy to show that it is increasing by looking at its derivative, so the function $f(x)=x^\frac{1}{x} - \frac{1}{\sqrt{x}}$ is increasing. Then what you need to do is to check the case where $n=3$.