Question

Use the Direct Comparison Test to determine the convergence of the given series; state what series is used for comparison. $$\sum_{n=1}^{\infty} \frac{\ln n}{n}$$

Short answer

The series diverges, compared with the divergent harmonic series.

Step-by-step solution

Understand the Convergence Criteria

The series given is \( \sum_{n=1}^{\infty} \frac{\ln n}{n} \). We need to determine whether this series converges. The Direct Comparison Test is a useful method to establish convergence by comparing the original series to a known convergent or divergent series.

Choose a Comparison Series

In order to use the Direct Comparison Test, we need to identify a series that we know is divergent or convergent. A good choice here is the harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \), which is a well-known divergent series.

Establish the Inequality for Comparison

For the Direct Comparison Test to be used, we need to compare \( \frac{\ln n}{n} \) with \( \frac{1}{n} \). Notice that for all \( n \geq 1 \), \( \ln n \geq 1 \). Therefore, \( \frac{\ln n}{n} \geq \frac{1}{n} \).

Apply the Direct Comparison Test

Since \( \frac{\ln n}{n} \geq \frac{1}{n} \) and \( \sum_{n=1}^{\infty} \frac{1}{n} \) diverges, by the Direct Comparison Test, the series \( \sum_{n=1}^{\infty} \frac{\ln n}{n} \) also diverges.

Key concepts

Convergent Series

A convergent series is a sequence of numbers where the sum approaches a specific finite number as more terms are added. This means, as you keep adding terms indefinitely, instead of the sum growing larger or smaller without bound, it steadily approaches a particular value, often denoted as \( S \).
For a series \( \sum_{n=1}^{\infty} a_n \) to be convergent, the limit of the partial sum \( S_n \) as \( n \to \infty \) must equal \( S \):\[\lim_{n \to \infty} S_n = S.\]
Characteristics of a convergent series:

• The partial sums get closer and closer to a fixed number.
• It never "blows up" to infinity.
• You can often use tests like the Direct Comparison Test to determine convergence.

Convergent series are crucial in mathematics as they ensure stability and provide meaningful results in calculations involving infinite processes.

Divergent Series

Opposite to a convergent series, a divergent series is one where the sum does not settle towards any particular number as more terms are added. Instead, the sum may grow towards infinity, negative infinity, or oscillate indefinitely.
When dealing with the series \( \sum_{n=1}^{\infty} a_n \), if\[\lim_{n \to \infty} S_n eq \text{finite},\]
then the series diverges. Here's what makes divergent series interesting:

• The partial sums do not approach any single, finite value.
• Understanding divergence helps in identifying series that behave unstably over infinite terms.
• Some divergent series can be compared to others to determine their behavior, as shown with the Direct Comparison Test.

In the given problem, the series \( \sum_{n=1}^{\infty} \frac{\ln n}{n} \) diverges because it compares directly to the divergent harmonic series.

Harmonic Series

The harmonic series is one of the most famous examples of a divergent series. It is given by:\[\sum_{n=1}^{\infty} \frac{1}{n}.\]
Although it seems counterintuitive due to the terms getting smaller and smaller, the sum of the harmonic series actually grows without bound. Here's what makes the harmonic series special:

• Despite each term becoming tiny, their sum still diverges to infinity.
• The divergence of the harmonic series was a critical discovery in calculus and helped shape the study of infinite sums.
• It serves as a benchmark for comparing other series, as seen in the Direct Comparison Test.

In the exercise, the harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \) is used to demonstrate that \( \sum_{n=1}^{\infty} \frac{\ln n}{n} \) diverges, based on the comparison that \( \frac{\ln n}{n} \geq \frac{1}{n} \). Understanding the behavior of the harmonic series helps in grasping why increasing sums of such fractional terms do not converge.