Determine the convergence or divergence of the series. 1n(In n)3n=2

Consider the series ∑n=2∞1nlnn3

Answered: 1 n(In n)3 n=2

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter1: Fundamental Concepts Of Algebra

Section1.1: Real Numbers

Problem 51E

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Question

Determine the convergence or divergence of the series.

Expert Solution

Step 1

Consider the series

$\sum_{n = 2}^{\infty} \frac{1}{n {(\ln n)}^{3}}$

Step 2

Cauchy Condensation test:

Let $〈c_{n}〉$ be a decreasing sequence of positive real numbers. Then the two series $\sum_{n = 1}^{\infty} c_{n} and \sum_{n = 1}^{\infty} 2^{n} c_{2^{n}}$ either convergent or divergent.

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