Convergence of logarithmic series with Cauchy condensation formula

Question

the series is $$\frac{1}{n^{\log{\log{\log n}}}}$$ So I put Cauchy condensation formula and got $\frac{2^n}{2^{n\log \log \log2}}$ and then root test and found that the limit $u^{1/n}$ is $0$ and hence series is convergent. I want to know if what I've done is correct? Thank you