Basic Analysis I — Real Analysis · Problem 2.5.16

Sequences and Series — Jiri Lebl

Mathematics — Real Analysis · cauchy-condensation

Problem

Use the Cauchy condensation principle (see Exercise 2.5.15) to decide the convergence of a)  ∑n=1∞ln⁡nn2b)  ∑n=2∞1nln⁡nc)  ∑n=2∞1n(ln⁡n)2d)  ∑n=2∞1n(ln⁡n)(ln⁡ln⁡n)2a)\; \sum_{n=1}^{\infty} \frac{\ln n}{n^2} \qquad b)\; \sum_{n=2}^{\infty} \frac{1}{n \ln n} \qquad c)\; \sum_{n=2}^{\infty} \frac{1}{n(\ln n)^2} \qquad d)\; \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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