TITLE: Show the series $\sum (2^{1/n}-1)$ diverges
QUESTION [8 upvotes]: How to show that the series
$$ \sum_{n=1}^\infty (\sqrt[n]{2}-1)$$
diverges ?

REPLY [2 votes]: $\lim_{x \to 0}\frac{2^x-1}{x} = \log2$. This implies $\lim_{n \to +\infty} \frac{2^\frac{1}{n}-1}{\frac{1}{n}} = \log2$. Since the series $\sum_{n=1}^{+\infty} \frac{1}{n}$ is divergent, so by Comparison test (limit form), the series $\sum_{n=1}^{+\infty} (\sqrt[n]{2}-1)$ is divergent.