# Test the convergence of $\sum_{n=3}^{\infty}\frac{1}{(\log n)^{\log\log n}}$ [duplicate]

I know that $$\sum_{n=3}^{\infty}\frac{1}{(\log n)^{\log n}}$$ converges and hence applying the Comparison test, I get the ratio $$\frac{a_n}{b_n}=\frac{(\log n)^{\log n}}{(\log n)^{\log\log n}}.$$ What I want to know is whether $a_n/b_n$ is bounded above or not. Or is there different approach to solving this problem?

• Cauchy's condensation test readily gives the answer. May 22 '17 at 15:55