Problem 12 - Thomas Calculus

Question

Determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test. $$ \sum_{n=1}^{\infty}(-1)^{n} \ln \left(1+\frac{1}{n}\right) $$

Step-by-Step Solution

Answer

The series converges.

Step 1: Identify the Alternating Series

The series given is \( \sum_{n=1}^{\infty}(-1)^{n} \ln \left(1+\frac{1}{n}\right) \). It is an alternating series because each term changes sign, indicated by the \((-1)^{n}\) factor.

Step 2: Examine the Non-Alternating Part of the Series

The non-alternating part of the series is \( a_n = \ln \left(1+\frac{1}{n}\right) \). We need to determine if \( a_n \) is positive, decreasing, and if \( \lim_{n \to \infty} a_n = 0 \).

Step 3: Verify Positivity of \(a_n\)

For all \( n \geq 1 \), \( \ln \left(1+\frac{1}{n}\right) > 0 \) because \( 1 + \frac{1}{n} > 1 \), and the natural logarithm of a number greater than 1 is positive.

Step 4: Check Decreasing Nature of \(a_n\)

Consider the function \( f(x) = \ln \left(1+\frac{1}{x}\right) \). The derivative, \( f'(x) = -\frac{1}{x(x+1)} \), is negative for \( x > 0 \). Thus, \( f(x) \) is decreasing, and \( a_n \) is decreasing for \( n \geq 1 \).

Step 5: Check the Limit Condition

We find that \( \lim_{n \to \infty} a_n = \lim_{n \to \infty} \ln \left(1+\frac{1}{n}\right) = \ln(1) = 0 \). This satisfies the limit condition required by the Alternating Series Test.

Step 6: Conclusion Using the Alternating Series Test

The series \( \sum_{n=1}^{\infty}(-1)^{n} \ln \left(1+\frac{1}{n}\right) \) satisfies all conditions of the Alternating Series Test: \( a_n > 0 \), \( a_n \) is decreasing, and \( \lim_{n \to \infty} a_n = 0 \). Therefore, the series converges.

Key Concepts

Understanding Series Convergence

Series convergence is a fundamental concept in mathematics, especially when dealing with infinite series. A series converges if the sum of its terms approaches a fixed number as more terms are added.
In the context of alternating series, which are series that alternate in signs, the Alternating Series Test becomes useful. This test examines whether a particular alternating series is convergent.
For a series to satisfy the Alternating Series Test, it must meet these criteria:

• The terms of the series must be positive when ignoring the (-1) multiplier.
• The sequence of terms must be decreasing.
• The limit of the terms must approach zero as they extend to infinity.

By confirming these conditions, we determine that the given series \[\sum_{n=1}^{\infty}(-1)^{n} \ln \left(1+\frac{1}{n}\right) \] converges. The challenge with this series is determining the behavior of \( a_n = \ln \left(1+\frac{1}{n}\right) \), which requires a look at its properties and their compliance with the above criteria.

Exploring Logarithmic Functions

Logarithmic functions, often involving the natural logarithm \( \ln(x) \), are crucial in analyzing series with logarithmic terms. In our series, \(\ln \left(1+\frac{1}{n}\right)\) depicts a common logarithmic expression where \(n\) is a positive integer.
Here’s why it's relevant:

• Logarithmic functions are known to grow slowly and decrease smoothly. This property works favorably when checking the reduction of terms in a series.
• For \( n \geq 1 \), \( 1 + \frac{1}{n} > 1 \), leading to a positive outcome from \( \ln(x) \).
• The derivative \( f'(x) = -\frac{1}{x(x+1)} \) reveals that \( f(x) = \ln \left(1+\frac{1}{x}\right) \) is decreasing, which impacts our series positively in verifying the Alternating Series Test conditions.

Understanding these properties helps in verifying that our series terms indeed decrease as required.

Applying Mathematical Proof

Mathematical proof is essential for verifying that series such as ours meet the convergence criteria. To conclude the convergence of \[\sum_{n=1}^{\infty}(-1)^{n} \ln \left(1+\frac{1}{n}\right) \]we used several analytical steps.

• We first identified the non-alternating part \(a_n = \ln \left(1+\frac{1}{n}\right)\) and examined its properties.
• The positivity was verified using the property of natural logarithms for values \(>1\).
• We calculated the derivative to confirm \(a_n\) is decreasing over time.
• Finally, by taking the limit \( \lim_{n \to \infty} a_n = 0 \), we secured the final condition for the Alternating Series Test.

These steps build a solid mathematical proof eluding the intuitive understanding that despite the series terms alternating signs, their diminishing size leads to an overall convergent series. Mathematical proofs are crucial in establishing faith in results and ensuring accuracy in higher mathematical endeavors.