Studying the character of $\sum_{n=3}^\infty \frac{1}{n(\log(\log n))^{\alpha}}$

Question

I have to study the character of this series $$\sum_{n=3}^\infty \frac{1}{n(\log(\log n))^{\alpha}}$$

with $\alpha$ a real parameter.

Considering the Cauchy condensation test, the equivalent series is: $$\sum_{n=1}^\infty 2^n\frac{1}{2^n[\log(\log 2^n)]^{\alpha}}=\sum_{n=1}^\infty \frac{1}{[\log(n\log 2)]^{\alpha}}$$

Using the ratio test:

$$\lim_{n\rightarrow \infty} \frac{[\log(n\log 2)]^{\alpha}}{[\log((n+1)\log 2)]^{\alpha}}=\lim_{n\rightarrow \infty} \frac{1}{[\log(\log 2)]^{\alpha}}= \frac{1}{[\log(\log 2)]^{\alpha}}\sim \frac{1}{(-0,36)^{\alpha}}$$

Then when $(-0,36)^{\alpha}>1$ the given series converges, otherwise it diverges

if $\alpha =1$ ,$ -0,36<1$ , diverges

if $\alpha =0$ ,$ 1<1$ , diverges

if $\alpha =-1$ ,$ (-0,36)^{-1}=-2,77<1$ , converges

if $\alpha >1$ ,$ (-0,36)^{\alpha}<1$ , converges

if $\alpha <-1$ ,$ (-0,36)^{\alpha}<1$ , converges

if $|\alpha| <1 , \ne 0$ sometimes it doen't exist $ (-0,36)^{\alpha}$ Can someone help me?

Answer 1

HINT

Let use limit comparison test with

$$\sum \frac{1}{n}$$

Answer 2

Since $\sum \dfrac1{n \log(n)}$ diverges and $\dfrac{\log n}{(\log \log n)^a} \to \infty$ for $a > 0$, the sum diverges for all $a$ since $\dfrac1{n(\log \log n)^a} =\dfrac1{n\log n}\dfrac{\log n}{(\log \log n)^a} $.

Answer 3

You've made a mistake using the ratio test. The ratio of logarithms isn't the logarithm of a difference, it's the other way around (the logarithm of a ratio is the difference of logarithms). Correctly you have: \begin{align} \lim_{n\rightarrow\infty} \frac{\big(\log(n\log 2)\big)^\alpha }{\big(\log((n+1)\log 2)\big)^\alpha} &= \lim_{n\rightarrow\infty} \left(\frac{\log n +\log\log 2}{\log(n+1)+ \log\log 2}\right)^\alpha = \\ &= \lim_{n\rightarrow\infty} \left(\frac{\log n +\log\log 2}{\log n + \log(1+\frac{1}{n})+ \log\log 2}\right)^\alpha = \\ &= \lim_{n\rightarrow\infty} \left(\frac{1 + \frac{\log\log 2}{\log n}}{1 +\frac{\log(1+\frac{1}{n})+ \log\log 2}{\log n}}\right)^\alpha = \\ &= \left(\frac{1 + 0}{1 +0}\right)^\alpha = 1\end{align} which means that the result is inconclusive. You need a different test, for example a comparison test with $\sum\frac{1}{n}$, or another Cauchy condensation test.