The OP asks the question [now deleted] whether the series $S =\sum_{k=2}^{\infty}(k^{1/k} -1)$ diverges. 

The answer is that the series diverges, because $$k^{\frac{1}{k}}-1 = \exp\left(\frac{\ln k}{k}\right)-1>\frac{\ln k}{k} > \frac{1}{k}$$
and $\sum_{k=2}^\infty 1/k=\infty$.