Problem 38 - Essential Calculus Early Transcendentals

Question

We have seen that the harmonic series is a divergent series whose terms approach 0. Show that $$\sum_{n=1}^{\infty} \ln \left(1+\frac{1}{n}\right)$$ is another series with this property.

Step-by-Step Solution

Answer

The series \( \sum_{n=1}^{\infty} \ln \left(1+\frac{1}{n}\right) \) diverges by comparison to the harmonic series.

Step 1: Analyze the Given Series

Consider the series \( \sum_{n=1}^{\infty} \ln \left(1+\frac{1}{n}\right) \). Each term of the series is \( \ln \left(1+\frac{1}{n}\right) \). Our goal is to show that this series diverges.

Step 2: Simplify a Term Using Logarithm Properties

Using the property of logarithms, \( \ln(a) - \ln(b) = \ln(\frac{a}{b}) \), we can rewrite \( \ln \left(1+\frac{1}{n}\right) = \ln \left(\frac{n+1}{n}\right) \).

Step 3: Expand the Logarithm Using Series Approximation

For any small \( x \), \( \ln(1+x) \approx x \), so \( \ln \left(1+\frac{1}{n}\right) \approx \frac{1}{n} \). This approximation suggests that the series behaves similarly to the harmonic series \( \sum \frac{1}{n} \), which is known to be divergent.

Step 4: Establish Divergence via Comparison Test

To confirm the approximation, notice that \( \ln(1+\frac{1}{n}) \) is greater than \( \frac{1}{2n} \). Since \( \sum \frac{1}{2n} = \frac{1}{2} \sum \frac{1}{n} \) is half of the harmonic series and diverges, by the Comparison Test the original series \( \sum \ln(1+\frac{1}{n}) \) diverges.

Key Concepts

Harmonic Series

The harmonic series is a fundamental concept in mathematics, particularly in series and sequences. It is defined as the infinite sum \( \sum_{n=1}^{fty} \frac{1}{n} \). At first glance, it might seem like it should converge because the terms \( \frac{1}{n} \) get smaller as \( n \) increases. However, the harmonic series is indeed a divergent series. This means that as you add more and more terms, the sum grows without bound, approaching infinity rather than settling down to a finite number. This characteristic of the harmonic series is important in various fields of mathematics and engineering.
Despite each term's diminishing contribution, the collective effect is significant enough to prevent the series from converging. This property of divergence is leveraged in comparing other similar series to determine their convergence or divergence. Understanding the nature of the harmonic series can greatly aid in grasping more complex ideas in analysis and beyond.

Comparison Test

The Comparison Test is a handy tool for determining whether a series converges or diverges. It is particularly useful when dealing with series that are difficult to analyze directly. The test involves comparing the terms of a series with those of a second series whose convergence properties are already known. There are two main versions of the Comparison Test:

• Direct Comparison Test: If the terms of your series are smaller than the terms of a known convergent series, then your series converges. Conversely, if the terms are larger than a known divergent series, your series diverges.
• Limit Comparison Test: This involves taking the limit of the ratio of the terms of two series. If the limit is a positive number, then both series either converge or diverge together.

In the exercise, the Comparison Test was used to show that the series \( \sum \ln(1 + \frac{1}{n}) \) diverges. By comparing it to a fraction of the harmonic series, it was proven that since one series diverges, so does the other. This method is very efficient when handling series that resemble or are fundamentally linked to simple, well-understood series like the harmonic series.

Logarithm Properties

Logarithms have several properties that facilitate the simplification and manipulation of mathematical expressions involving them. These properties can be very useful in calculus and series analysis. Here are some key logarithm properties used in the exercise:

• Quotient Rule: \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \) helps in breaking down complex logarithmic terms into simpler parts.
• Approximation for Small \(x\): When \(x\) is small, \(\ln(1 + x) \approx x\). This approximation is key to examining series, especially in proving divergence or convergence, by relating logarithmic series to more basic series like the harmonic series.

In the original step by step solution, logarithm properties were essential in transforming and approximating the series term \( \ln(1 + \frac{1}{n}) \) into a format that shows its similarity to terms of the harmonic series. Recognizing and applying these properties ease the analysis of series and logarithmic functions across various disciplines.