Use the nth-Term Test for divergence to show that the series is divergent, or state that the test is inconclusive.
-$$\sum _ { n = 1 } ^ { \infty } \ln \frac { 6 } { n }$$
A) diverges
B) converges; ln 6
C) inconclusive
D) converges; ln 16

Question

Quizplus · Accepted Answer

The nth-Term Test for divergence states that if the limit of the nth term of a series does not approach zero as n goes to infinity, then the series diverges.

In this case, the nth term of the series is simply ln(6/n). We can evaluate the limit of this term as n approaches infinity using L'Hôpital's rule:

lim (n → ∞) ln(6/n) = lim (n → ∞) [ln(6) - ln(n)] 
= ln(6) - lim (n → ∞) ln(n)

The limit of ln(n) as n approaches infinity is infinity, so the limit of the nth term is negative infinity. Since the limit does not approach zero, the series diverges. Thus, the answer is A) diverges.

Use the Nth-Term Test for Divergence to Show That the Series