Is the series $\sum\limits_{k=1}^{\infty}( a^{\frac{1}{k}}-1)$ convergent?

Question

Assume that $a \gt 1$ .

Is the series $\sum\limits_{k=1}^{\infty}( a^{\frac{1}{k}}-1)$ convergent?

Note : I think it must be related to a test ... Is there a test to confirm that a series converges ? Can we say this series converges because $a^{\frac{1}{k}}$ does?

Answer 1

Hint: $$a^{1/k}-1=\int_0^{1/k}a^x\ln a\,dx\sim\frac{\ln a}{k}$$

Answer 2

Hint: $$a^{1/k} - 1 = e^{(\log a)/k} -1\approx\frac{\log a}k$$