Problem 33 - Thomas Calculus

Question

In Exercises \(27-34,\) use the \(n\) th-Term Test for divergence to show that the series is divergent, or state that the test is inconclusive. $$ \sum_{n=1}^{\infty} \ln \frac{1}{n} $$

Step-by-Step Solution

Answer

The series diverges by the nth-Term Test because \(\lim_{{n \to \infty}} -\ln(n) = -\infty\).

Step 1: Determine the nth-Term of the Series

Identify the general term of the series, which is \(a_n = \ln \frac{1}{n}\).

Step 2: Simplify the nth-Term Formula

Rewrite the term using properties of logarithms: \(a_n = \ln(1) - \ln(n) = -\ln(n)\).

Step 3: Evaluate the Limit of the nth-Term

To apply the nth-Term Test for divergence, calculate \(\lim_{{n \to \infty}} a_n = \lim_{{n \to \infty}} -\ln(n)\).

Step 4: Analyze the Limit Behavior

As \(n\) approaches infinity, \( -\ln(n) \to -\infty\). Therefore, \(\lim_{{n \to \infty}} a_n = -\infty\).

Step 5: Apply the nth-Term Test

The nth-Term Test states that if \(\lim_{{n \to \infty}} a_n eq 0\), then the series \(\sum a_n\) diverges. Since \(-\infty \) is not zero, the series diverges.

Key Concepts

nth-Term Test

The nth-Term Test is a simple yet powerful tool to determine whether an infinite series diverges. When you have a series such as \( \sum_{n=1}^{\infty} a_n \), this test checks the limit of the general term \( a_n \) as \( n \) approaches infinity. The key idea is that if this limit is not zero, the series cannot converge to a sum, and hence, it diverges.

Here's how it works step-by-step:

• Identify the general term \( a_n \) of the series.
• Compute \( \lim_{{n \to \infty}} a_n \).
• If the limit is not zero, the series diverges. If the limit is zero, the test is inconclusive, and further tests are needed.

For our series, \( a_n = -\ln(n) \), the limit as \( n \to \infty \) is \(-\infty \). Since this is not zero, the series diverges according to the nth-Term Test.

infinite series

An infinite series is a sum of infinitely many terms. It's written in the form \( \sum_{n=1}^{\infty} a_n \).

Infinite series can describe a wide variety of numerical and functional behaviors. They can either converge or diverge:

• A convergent series sums to a finite value, while a divergent series does not.
• There are various tests like the nth-Term Test, the Ratio Test, and the Integral Test to determine if a series converges or diverges.

For series like \( \sum_{n=1}^{\infty} -\ln(n) \), which are represented by continuously increasing negative values, convergence is hard to achieve since they tend to become unbounded or infinite. The nth-Term Test we just discussed confirms that such a series diverges.

logarithm properties

Logarithm properties are essential for simplifying expressions involving logarithms like \( \ln \frac{1}{n} \).

Key logarithm properties include:

• Product Property: \( \ln(ab) = \ln(a) + \ln(b) \)
• Quotient Property: \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \)
• Power Property: \( \ln(a^b) = b \cdot \ln(a) \)

For the series from the exercise, we used the quotient property to rewrite \( \ln \frac{1}{n} \) as \( \ln(1) - \ln(n) \), simplifying it to \(-\ln(n)\). Understanding these properties allows you to manipulate logarithmic expressions effectively, which is crucial for both solving mathematical problems and conducting deeper analysis.