Which of the following best justifies the convergence of the series ?

MATHEMATICS · Single correct · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Which of the following best justifies the convergence of the series ∑n=2∞1n(ln⁡n)2\displaystyle \sum_{n=2}^{\infty} \frac{1}{n(\ln n)^2}? (a) Because lim⁡n→∞1n(ln⁡n)2=0\displaystyle\lim_{n\to\infty}\dfrac{1}{n(\ln n)^2}=0, the series must converge. (b) For all n≥2n\ge2, 1n(ln⁡n)2≤1n\dfrac{1}{n(\ln n)^2}\le\dfrac{1}{n}, and since ∑n=2∞1n\sum_{n=2}^{\infty}\dfrac{1}{n} diverges, the given series diverges by comparison. (c) For sufficiently large nn, 1n(ln⁡n)2≤1n1.1\dfrac{1}{n(\ln n)^2}\le\dfrac{1}{n^{1.1}}, and because ∑n=2∞1n1.1\sum_{n=2}^{\infty}\dfrac{1}{n^{1.1}} converges (p>1)(p>1), the given series also converges by comparison. (d) Using the Ratio Test, lim⁡n→∞an+1an=12<1\displaystyle\lim_{n\to\infty}\dfrac{a_{n+1}}{a_n}=\tfrac12<1, so the series converges.

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