Chapter 8.4, Problem 59E

Loglog p-series Consider the series ${\displaystyle \underset{k=3}{\overset{\ne }{\sum }}\frac{1}{k(\ln k){(\ln \ln k)}^p}}$, where p is a real number.

1. a. For what values of p does this series converge?
2. b. Which of the following series converges faster? Explain.

${\displaystyle \underset{k=2}{\overset{\ne }{\sum }}\frac{1}{k{(\ln k)}^p}}\text{or}{\displaystyle \underset{k=3}{\overset{\ne }{\sum }}\frac{1}{k(\ln k){(\ln \ln k)}^p}?}$