Question

State whether each of the following series converges absolutely, conditionally, or not at all\(\sum_{n=1}^{\infty}(-1)^{n+1}(\ln (n+1)-\ln n)\)

Short answer

The series converges conditionally.

Step-by-step solution

Understand the Series

The series given is \(\sum_{n=1}^{\infty}(-1)^{n+1}(\ln (n+1)-\ln n)\). This is an alternating series because the term \((-1)^{n+1}\) causes the terms in the series to alternate signs.

Simplify the Series Term

Each term in the series is \((-1)^{n+1}(\ln (n+1) - \ln n)\). Using the properties of logarithms, \(\ln (n+1) - \ln n = \ln\left(\frac{n+1}{n}\right)\). Thus, each term becomes \((-1)^{n+1}\ln\left(1 + \frac{1}{n}\right)\).

Check for Absolute Convergence

A series \(\sum a_n\) converges absolutely if \(\sum |a_n|\) converges. Here, \(|a_n| = \ln\left(1 + \frac{1}{n}\right)\). For large \(n\), \(\ln\left(1 + \frac{1}{n}\right)\) is approximately \(\frac{1}{n}\). The series \(\sum \frac{1}{n}\) is a harmonic series, which diverges. Thus, \(\sum |a_n|\) does not converge, so the series does not converge absolutely.

Check for Conditional Convergence Using the Alternating Series Test

For the alternating series test, the terms of the series \(a_n = \ln\left(1 + \frac{1}{n}\right)\) must satisfy two conditions: they must be positive, decreasing, and their limit as \(n\to\infty\) must be zero. Here, \(\ln\left(1 + \frac{1}{n}\right) > 0\) for all \(n\), \(\ln\left(1 + \frac{1}{n}\right)\) is decreasing as \(n\) increases, and \(\lim_{n\to\infty} \ln\left(1 + \frac{1}{n}\right) = 0\). So, by the alternating series test, the series converges.

Conclusion on the Type of Convergence

Since the series converges by the alternating series test but does not converge absolutely, it converges conditionally.

Key concepts

Alternating Series

An alternating series is a series where the terms change sign with each subsequent term. This means that after a positive term, a negative term follows, and then a positive term again. The pattern continues in this fashion.

Alternating series often have the form \[ a_1 - a_2 + a_3 - a_4 + ext{...} \]where the terms can be represented by \[ (-1)^{n+1} a_n \]signifying they alternate in sign. Alternating series are significant because they possess unique convergence properties. Notably, they can converge even when the sequence of absolute values, \[ |a_n| \]forms a divergent series.

With these series, the alternating series test can be applied, providing a method to determine whether a series converges. The test requires that the terms decrease in magnitude and approach zero as an index goes to infinity. If these conditions are met, the series converges.

Absolute Convergence

Absolute convergence refers to scenarios where summing the absolute values of a series' terms results in a convergent series. Specifically, a series \( \sum a_n \) converges absolutely if \( \sum |a_n| \)is a convergent series. This condition is robust, because it implies that the series converges regardless of any positive or negative signs. This is stronger than just regular convergence.

If a series converges absolutely, it is guaranteed to be convergent; however, the reverse may not always hold true. Absolute convergence guarantees control over sign changes, making it a useful property in various analyses. For instance, absolute convergence prevents divergence due to fluctuations triggered by alternating sign terms.

Conditional Convergence

Conditional convergence happens when a series converges, but it does not converge absolutely. In simpler terms, a series is said to be conditionally convergent if it converges when its terms are taken with their original signs, but it does not converge when all terms are made positive by taking the absolute value.

Such series highlight an interesting aspect of convergence, as they depend on the arrangement of terms. A classic example is the alternating harmonic series, represented as:\[ 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots \]which converges conditionally. Examining conditional convergence requires tests like the alternating series test to confirm convergence of the original series without absolute values.

Harmonic Series

The harmonic series is a basic but important series in mathematics, defined as\[ \sum_{n=1}^{\infty} \frac{1}{n} \]It is called harmonic because of its connection to musical harmony and frequency ratios. In nature and engineering, the harmonic series often crops up, playing a critical role in various functions.

Interestingly, despite each added term getting smaller and smaller, the harmonic series diverges, meaning it grows indefinitely without bound. This divergence can be demonstrated formally or informally by showing that the partial sums grow larger than any fixed number as more terms are added. Thus, even though terms shrink towards zero, the series itself does not settle to a finite limit.

Logarithmic Properties

Logarithmic properties are useful tools in simplifying expressions and evaluating series like the one presented. These properties include common rules such as:

• \( \ln(ab) = \ln a + \ln b \)
• \( \ln\left(\frac{a}{b}\right) = \ln a - \ln b \)
• \( \ln(a^b) = b \ln a \)

For series involving logarithms, especially those alternating or conditionally convergent, these properties let us rewrite terms in a way that reveals patterns or simplifies calculation.

In the given series, by using \( \ln(\frac{n+1}{n}) \), we leverage the logarithmic difference property to simplify and analyze the series' convergence nature. This simplification shows the intricacies and underlying behaviors of finite parts within infinite series, which can significantly impact convergence assessment.