Question

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{1}{n}
select the correct choice below and, if necessary, fill in the answer box within your choice.
a. the series diverges because lim_{n
ightarrowinfty}ln\frac{1}{n} exists and is equal to
b. the series diverges because lim_{n
ightarrowinfty}ln\frac{1}{n}=infty and fails to exist.
c. the series diverges because lim_{n
ightarrowinfty}ln\frac{1}{n}=-infty and fails to exist.
d. the test is inconclusive because lim_{n
ightarrowinfty}ln\frac{1}{n}=

Explanation:

Step1: Recall the property of the natural logarithm

We know that \(\ln\frac{1}{n}=-\ln n\).

Step2: Find the limit as \(n\to\infty\)

We calculate \(\lim_{n\to\infty}\ln\frac{1}{n}=\lim_{n\to\infty}(-\ln n)\). As \(n\to\infty\), \(\ln n\to\infty\), so \(-\ln n\to-\infty\).

Answer:

C. The series diverges because \(\lim_{n\to\infty}\ln\frac{1}{n}=-\infty\) and fails to exist.