Investigate the series .

MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Investigate the series ∑n=3∞1  n ln⁡n ln⁡(ln⁡n)  \displaystyle\sum_{n=3}^{\infty}\dfrac{1}{\;n\,\ln n\,\ln(\ln n)\;}. (1) For x≥3x\ge 3, let h(x)=1x ln⁡x ln⁡(ln⁡x)h(x)=\dfrac{1}{x\,\ln x\,\ln(\ln x)}. Prove that hh is positive, continuous, and decreasing on [3,∞)[3,\infty). (2) Compute or otherwise evaluate the improper integral ∫3∞1x ln⁡x ln⁡(ln⁡x) dx\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. (3) 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. (4) Discuss how your conclusion would change, if at all, if the exponent of the innermost logarithm were pp, giving the general series ∑n=3∞1n ln⁡n [ln⁡(ln⁡n)]p\displaystyle\sum_{n=3}^{\infty}\dfrac{1}{n\,\ln n\,[\ln(\ln n)]^{p}}. Identify the values of pp for which the series converges and for which it diverges, supporting your claims with appropriate reasoning.

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