Proving the convergence/divergence of a seemingly oscillating series

1,088

Let $b_n = \frac{1}{\ln{(\ln{(n)})}}$

Since $\ln(n)$ is increasing, we know $\ln{(\ln{(n)})}$ also increases, thus we have that:

$b_n = \frac{1}{\ln{(\ln{(n)})}}$ is monotonically decreasing on $[2,\infty)$ and also $$\lim_{n\to \infty}b_n= \lim_{n\to \infty}\frac{1}{\ln{(\ln{(n)})}} = 0.$$ Thus, from Leibniz's Test for Alternating Series, we know $\displaystyle \sum_{n=3}^\infty \frac{(-1)^{n-1}}{\ln{(\ln{(n)})}}$ converges.

Share:
1,088
user3776574
Author by

user3776574

Updated on November 19, 2020

Comments

  • user3776574
    user3776574 almost 3 years

    How does one figure out whether this series: $$\sum_{n=3}^{\infty}(-1)^{n-1}\frac{1}{\ln\ln n}$$ converges or diverges? And, what is the general approach behind solving for convergence/divergence in a series that seems to "oscillate" (thanks to the -1 in this case)?

    I have so far tried to split the function into two limits, but I am more or less stuck there.