Problem 30 - Thomas Calculus

Question

Which of the series Converge absolutely, which converge, and which diverge? Give reasons for your answers. $$ \sum_{n=1}^{\infty}(-1)^{n} \frac{\ln n}{n-\ln n} $$

Step-by-Step Solution

Answer

The series converges conditionally.

Step 1: Understand the Series

The given series is \( \sum_{n=1}^{\infty}(-1)^{n} \frac{\ln n}{n-\ln n} \). This is an alternating series, as the factor \((-1)^{n}\) causes the terms to alternate in sign.

Step 2: Check Absolute Convergence

To check for absolute convergence, we consider the series of absolute values: \( \sum_{n=1}^{\infty} \left| \frac{\ln n}{n-\ln n} \right| \), which simplifies to \( \sum_{n=1}^{\infty} \frac{\ln n}{n-\ln n} \). If this series converges, then the original series converges absolutely.

Step 3: Apply the Ratio Test for Absolute Convergence

Apply the ratio test to the series \( \sum_{n=1}^{\infty} \frac{\ln n}{n-\ln n} \).Compute \( \lim_{n \to \infty} \frac{|a_{n+1}|}{|a_n|} \) where \( a_n = \frac{\ln n}{n-\ln n} \).Evaluate \( \frac{\ln(n+1)}{(n+1)-\ln(n+1)} \times \frac{n-\ln n}{\ln n} \).Simplifying yields \( \lim_{n \to \infty} \frac{\ln(n+1)(n-\ln n)}{(n+1-\ln(n+1))\ln n} \to 1 \).Since the limit is 1 and not less than 1, the test is inconclusive for absolute convergence.

Step 4: Check Conditional Convergence with the Alternating Series Test

For an alternating series \( \sum (-1)^n a_n \), if \( a_n > 0 \), \( a_n \to 0 \) and \( a_{n+1} \leq a_n \) for all \( n \), the series converges.Here, \( a_n = \frac{\ln n}{n-\ln n} \). As \( n \to \infty \), \( \frac{\ln n}{n-\ln n} \to 0 \) because \( n-\ln n \approx n \). Check if \( a_{n+1} \leq a_n \):This is true as the denominator increases slightly faster than the numerator, satisfying \( a_{n+1} \leq a_n \).Since both conditions are met, the alternating series converges conditionally.

Step 5: Conclusion

The series \( \sum_{n=1}^{\infty} (-1)^n \frac{\ln n}{n-\ln n} \) does not converge absolutely. However, it converges conditionally by the Alternating Series Test.

Key Concepts

Convergence Tests

Convergence tests are essential tools in determining whether a given series converges or diverges. In our exercise, we have an alternating series, denoted by the factor \((-1)^n\), which causes the terms to switch signs with each new term. The Ratio Test is one method to test for absolute convergence. For this method, we look at the limit as \(n\) approaches infinity of the absolute value of the ratio of consecutive terms. Mathematically, this involves computing \(\lim_{n \to \infty} \frac{|a_{n+1}|}{|a_n|}\). If this limit is less than 1, the series converges absolutely. If the limit is greater than 1, it diverges. And if the limit equals 1, the test is inconclusive. In our case, the limit was found to be 1, which means we couldn't draw a firm conclusion about absolute convergence from the Ratio Test alone. Instead, we then turned to the Alternating Series Test to investigate convergence, achieving a conclusive result for conditional convergence.

Conditional Convergence

Conditional convergence occurs when a series converges, but does not converge absolutely. This can typically be tested using the Alternating Series Test. For an alternating series like \(\sum (-1)^n a_n\), the test involves checking two conditions:

• Each term \(a_n > 0\).
• The limit of \(a_n\) as \(n\) approaches infinity is zero (i.e., \(a_n \to 0\)).
• The sequence \(a_n\) is decreasing, meaning \(a_{n+1} \leq a_n\).

If these conditions are satisfied, the series converges conditionally. In our series, \(a_n = \frac{\ln n}{n-\ln n}\), which satisfies all the conditions listed. Thus, while the series does not converge absolutely (since the Ratio Test was inconclusive), it converges conditionally as shown by the Alternating Series Test.

Absolute Convergence

Absolute convergence is a stronger form of convergence and is implied when a series converges even when all terms are made non-negative (in absolute value). Checking for this involves forming a series of absolute values and determining its convergence. We attempted to verify absolute convergence in our example by applying the Ratio Test to the series \(\sum_{n=1}^{\infty} \left| \frac{\ln n}{n-\ln n} \right|\). Absolute convergence would imply that the original series also converges. However, the Ratio Test gave a limit of 1, which left the question of absolute convergence unresolved. Since this test was inconclusive, absolute convergence could not be established, meaning our series only converges conditionally.