Lebl-Basic-Analysis-I · Problem 2.5.16

Sequences and Series

Mathematics · cauchy-condensation

Problem

Use the Cauchy condensation principle (see Exercise 2.5.15) to decide the convergence of a) ∑n=1∞ln⁡nn2\displaystyle\sum_{n=1}^{\infty} \frac{\ln n}{n^2} b) ∑n=2∞1nln⁡n\displaystyle\sum_{n=2}^{\infty} \frac{1}{n \ln n} c) ∑n=2∞1n(ln⁡n)2\displaystyle\sum_{n=2}^{\infty} \frac{1}{n(\ln n)^2} d) ∑n=2∞1n(ln⁡n)(ln⁡ln⁡n)2\displaystyle\sum_{n=2}^{\infty} \frac{1}{n(\ln n)(\ln \ln n)^2} Note that only the tails of some of these series satisfy the hypotheses of the principle; you should argue why that is sufficient. *Hint: Feel free to use the identity ln⁡(2n)=nln⁡2\ln(2^n) = n \ln 2.*

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