How would one prove that the series
$$\sum_{n=1}^{\infty}(\sqrt[n]{e}-1)$$
diverges? The Root and Ratio test are useless here. It is also not so obvious to me how I could bound the given series by a smaller series which diverges.
A friend a mine told me that I can use the Cauchy Condensation Test and I got to this point
$$\sum_{n=1}^{\infty}(\sqrt[n]{e}-1) \sim \sum_{n=1}^{\infty}2^n(\sqrt[2n]{e}-1)$$
but it is unclear to me how to continue from here.