Mathematics

Problem

A student computes

\displaystyle\lim_{n\to\infty} \frac{\ln n}{n}=0

and therefore claims that the series

\sum_{n=1}^{\infty} \frac{\ln n}{n}

converges. Which statement best explains why the student’s conclusion is not justified? a) The

n

th-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

n

th-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

\dfrac{\ln n}{n}

\dfrac{1}{n^{2}}

instead; that comparison would decide convergence.

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