Question

Determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \ln \frac{1}{n} $$

Short answer

The series diverges according to the test for divergence since the sequence of terms \( \ln \left(\frac{1}{n}\right) \) does not approach zero as \( n \) goes to infinity.

Step-by-step solution

Identify the sequence of terms

The first thing to do is to identify the sequence of terms in the series which is \( a_n = \ln \left(\frac{1}{n}\right) \) for \( n \geq 1 \)

Apply the test for divergence

Next, determine whether the sequence of terms approaches zero as \( n \) approaches infinity, that is \( \lim_{n\to\infty} a_n =? \). Evaluate the limit using the properties of logarithms \( \lim_{n\to\infty} \ln \left(\frac{1}{n}\right) = \ln \left(\lim_{n\to\infty} \frac{1}{n}\right) \). Since \( \frac{1}{n} \) approaches zero as \( n \) goes to infinity, the limit is \( \ln(0) \). However, the natural logarithm of zero is undefined.

Determine convergence or divergence

Since the sequence of terms does not approach zero, the test for divergence indicates that the series diverges.