A student computes and therefore claims that the series converges

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

Problem

A student computes lim⁡n→∞ln⁡nn=0\displaystyle\lim_{n\to\infty} \frac{\ln n}{n}=0 and therefore claims that the series ∑n=1∞ln⁡nn\sum_{n=1}^{\infty} \frac{\ln n}{n} converges. Which statement best explains why the student’s conclusion is not justified? (a) The nnth-term test can be used only for series whose general terms contain exponentials, so it is not applicable here. (b) Because the limit of the terms is zero, the series actually diverges by the nnth-term test. (c) A limit of zero only shows that the test is inconclusive; additional convergence tests are needed. (d) The student should have compared ln⁡nn\dfrac{\ln n}{n} to 1n2\dfrac{1}{n^{2}} instead; that comparison would decide convergence.

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