Mathematics

Problem

Investigate the series

\displaystyle\sum_{n=3}^{\infty}\dfrac{1}{\;n\,\ln n\,\ln(\ln n)\;}

. a. For

x\ge 3

, let

h(x)=\dfrac{1}{x\,\ln x\,\ln(\ln x)}

. Prove that

h

is positive, continuous, and decreasing on

[3,\infty)

. b. Compute or otherwise evaluate the improper integral

\displaystyle\int_{3}^{\infty}\dfrac{1}{x\,\ln x\,\ln(\ln x)}\,dx

. Provide a rigorous derivation, clearly indicating any substitutions and limiting processes used. c. From your analysis in parts (a) and (b), decide whether the original series converges or diverges. Write a complete argument that references all relevant findings. d. Discuss how your conclusion would change, if at all, if the exponent of the innermost logarithm were

p

, giving the general series

\displaystyle\sum_{n=3}^{\infty}\dfrac{1}{n\,\ln n\,[\ln(\ln n)]^{p}}

. Identify the values of

p

for which the series converges and for which it diverges, supporting your claims with appropriate reasoning.

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