Problem 64 - Essential Calculus

Question

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

Step-by-Step Solution

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 1: 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 \( n \geq 1 \)

Step 2: 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.

Step 3: Determine convergence or divergence

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

Key Concepts

Divergence Test

The Divergence Test is a simple and essential tool in determining whether an infinite series diverges. For a series \( \sum a_n \) to converge, it is necessary for the sequence of terms \( a_n \) to approach zero as the index \( n \) approaches infinity. If the sequence does not tend to zero, the series must diverge. This is often the first test applied when assessing the convergence of a series.

To use the divergence test:

• Identify the sequence of terms \( a_n \) of the series.
• Compute \( \lim_{n\to\infty} a_n \)
• If \( \lim_{n\to\infty} a_n eq 0 \), the series diverges.
• If \( \lim_{n\to\infty} a_n = 0 \), the test is inconclusive.

In our exercise, the terms are \( a_n = \ln \left(\frac{1}{n}\right) \). As \( n \to \infty \), \( \frac{1}{n} \to 0 \), and since \( \ln(0) \) is undefined, the terms do not approach zero. Therefore, the series diverges.

Natural Logarithms

Natural logarithms (ln) are logarithms to the base \( e \), where \( e \) is an irrational constant approximately equal to 2.718. The function \( \ln(x) \) describes the power to which \( e \) must be raised to obtain the number \( x \). Natural logarithms are widely applicable in mathematical expressions, particularly in calculus and analysis.

Some key properties of natural logarithms include:

• The domain is \( (0, \infty) \), meaning \( x \) must be positive.
• \( \ln(1) = 0 \), since \( e^0 = 1 \).
• \( \ln(ab) = \ln(a) + \ln(b) \) - the product rule.
• \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \) - the quotient rule.
• \( \ln(a^b) = b \ln(a) \) - the power rule.

In our exercise, the term \( \ln\left(\frac{1}{n}\right) \) involves the logarithmic function applied to a fraction. As \( n \) increases, the argument of the logarithm \( \frac{1}{n} \) tends towards zero, highlighting the properties and behavior of natural logs.

Infinite Series

An infinite series is a sum of an infinite sequence of terms \( a_1, a_2, a_3, \ldots \) continuing indefinitely. Infinite series are a fundamental concept in mathematics, especially in calculus and analysis, used to describe many natural and theoretical phenomena.

There are different types of series:

• Convergent series: The sum approaches a specific number as more terms are added.
• Divergent series: The sum either increases without bound or oscillates without approaching any value.
• Conditionally convergent series: Converges only under particular arrangements of terms.

To determine the behavior of a series:

• Apply tests like the divergence test, which checks if terms go to zero.
• Sequence of partial sums, which helps visualize convergence or divergence.
• Techniques such as the ratio test, root test, or comparison test for more complex assessment.

In our scenario, the series \( \sum_{n=1}^{\infty} \ln \frac{1}{n} \) diverges as its sequence of terms does not fulfill the requirement of approaching zero.