Check convergence of $\sum^{\infty}_{n=1} \frac{1}{(\ln\ln n)^{\ln n}}$

Question

Check convergence of

$$\sum^{\infty}_{n=1}\frac{1}{(\ln \ln n)^{\ln n}}.$$

Please verify my solution below.

Answer 1

$$\sum^{\infty}_{n=1}\frac{1}{(\ln\, \ln\, n)^{\ln\, n}}=\sum^{\infty}_{n=1}\frac{1}{\exp( \ln\, \ln\ln\, n\,*\ln\,, n)}=\sum^{\infty}_{n=1}\frac{1}{\exp(\ln\, n\,*\ln\, \ln\, \ln\, n)}=\sum^{\infty}_{n=1}\frac{1}{n^{\ln\, \ln\, \ln\, n}}$$

For a lagre n, $$\frac{1}{n^{\ln\, \ln\, \ln\, n}} < \frac{1}{n^2}$$

So the series converge by comparison test

Answer 2

Use the inequality $$ \frac{x-1}{x^p}<\ldots<\frac{x-1}{x^2}<\frac{x-1}{x}<\log x , \quad \forall \,x>1, \,\forall p\in\mathbb{N}\backslash\{1,0\}. $$ and the fact that $\ln$ is an increasing function. We have for $n$ large, \begin{align} \left|\frac{1}{\left[\log\big(\log n\big)\right]^{(\log n)}}\right|\leq & \frac{1}{\left|\log\circ\log n\right|^{\big(\frac{n-1}{n^p}\big)}} \\ \leq & \frac{1}{\left|\log\big(\frac{n-1}{n}\big)\right|^{\big(\frac{n-1}{n^p}\big)}} \\ \leq & \frac{1}{\left|\frac{\big(\frac{n-1}{n}\big)-1}{\big(\frac{n-1}{n}\big)}\right|^{\big(\frac{n-1}{n^p}\big)}} \\ = & \frac{1}{\left(\frac{n}{n-1}\right)^{\big(\frac{n-1}{n^p}\big)}} \\ = & \bigg(1-\frac{1}{n}\bigg)^{\frac{n-1}{n^p}} \\ = & \left[\bigg(1-\frac{1}{n}\bigg)^{n}\right]^{\frac{1}{n^p}} \\ \end{align} And by root test, if $a_n= \left[\bigg(1-\frac{1}{n}\bigg)^{n}\right]^{\frac{1}{n^p}}$ we have $ \lim_{n\to \infty}\sqrt[n]{a_n}<1 $ for $p$ enogh large. By comparation test the serie $$ \sum_{n=1}^{\infty}\frac{1}{\left[\log\big(\log n\big)\right]^{(\log n)}} $$ converge.

Answer 3

There is $n_0$ such that

$$\sum^{\infty}_{n=n_0}\frac{1}{(\ln \ln n)^{\ln n}}<\sum^{\infty}_{n=n_0}\frac{1}{3^{\ln n}}=\sum^{\infty}_{n=n_0}\frac{1}{n^{\ln 3}}$$ and this converges since $\ln 3>1$

Q.E.D.