Is $sum_nleft(a_n^{frac{1}{n}}-1right)$ convergent?

Let ${a_n}$ is a sequence of positive real numbers and $limlimits_{nrightarrow infty}a_n^{frac{1}{n}}=1$ , then is there any condition on ${a_n}$ so that the $sum_nleft(a_n^{frac{1}{n}}-1right)$ is convergent $?$

I have seen one related question that : Prove or disprove that $sum_n left( n^{frac{1}{n}}-1right)$ is convergent.

My Approach is : $limlimits_{n rightarrow infty} left( 1+frac{1}{n}right)^n=e$, so $left( 1+frac{1}{n}right)^n<n$ for all $ngeq 3$. Hence $frac{1}{n} <n^{frac{1}{n}}-1$ for all $ngeq 3$. So by comparison test the given series is divergent.

But I can not figure out the above question.