Examine the convergence of $\sum_{n=1}^{\infty}\frac{1}{\sum_{k=1}^{n}k^{1/k}}$

Question

I know that $$\sum_{n=1}^{\infty}{n^{1/n}}$$ diverges, since the limit when $n$ tends to infinity of $n^\frac{1}{n}$ is equal to $1$. With this, I cannot conclude anything about the convergence of $$\sum_{n=1}^{\infty}\frac{1}{\sum_{k=1}^{n}k^{1/k}}$$ I don't see how to apply the root test, and I don't know which series I should choose to apply the comparison test. If I apply the Cauchy Condensation test we have $$\sum_{n=1}^{\infty}\frac{2^n}{\sum_{k=1}^{2^n}k^{1/k}}$$ and i don't know how to solve that limit