Problem 19 - Single Variable Calculus

Question

Determine whether the series is convergent, absolutely convergent, conditionally convergent, or divergent. \(\sum_{n=2}^{\infty} \frac{(-1)^{n} \ln n}{2^{n}}\)

Step-by-Step Solution

Answer

The given series \(\sum_{n=2}^{\infty} \frac{(-1)^{n} \ln n}{2^{n}}\) is absolutely convergent, as determined by applying the Ratio Test on its absolute value series and obtaining a limit of \(\frac{1}{2}<1\).

Step 1: Check for absolute convergence

First, we'll examine absolute convergence by taking the absolute value of the series terms and see if that series converges: \[\sum_{n=2}^{\infty} \left| \frac{(-1)^{n} \ln n}{2^{n}} \right| = \sum_{n=2}^{\infty} \frac{\ln n}{2^{n}}\]

Step 2: Ratio Test for Absolute Convergence

To determine the convergence of the absolute value series, we use the Ratio Test. For that, we'll compute the limit of the ratio of the terms: \[\lim_{n\to\infty} \frac{\left|\frac{\ln (n+1)}{2^{n+1}}\right|}{\left|\frac{\ln n}{2^n}\right|} = \lim_{n\to\infty} \frac{\ln (n+1)}{\ln n} \cdot \frac{2^n}{2^{n+1}}\]

Step 3: Simplify the Limit Expression

Now, we'll simplify the limit expression: \[\lim_{n\to\infty} \frac{\ln (n+1)}{\ln n} \cdot \frac{2^n}{2^{n+1}} = \lim_{n\to\infty} \frac{\ln (n+1)}{\ln n} \cdot \frac{1}{2}\]

Step 4: Evaluate the Limit

Using the rule \(\lim_{n\to\infty} \frac{\ln (n+1)}{\ln n} = 1\), we evaluate the limit: \[\lim_{n\to\infty} \frac{\ln (n+1)}{\ln n} \cdot \frac{1}{2} = 1 \cdot \frac{1}{2} = \frac{1}{2}\] Since the limit \(L = \frac{1}{2} < 1\), the absolute value series converges by the Ratio Test.

Step 5: Conclusion

Since the absolute value series converges, we can conclude that the original series is absolutely convergent. Therefore, there's no need to check for conditional convergence.

Step 6: Final Answer

The given series \(\sum_{n=2}^{\infty} \frac{(-1)^{n} \ln n}{2^{n}}\) is absolutely convergent.

Key Concepts

Absolute Convergence

In the realm of series and sequences, absolute convergence plays a pivotal role in understanding the behavior of infinite series. An infinite series \(\sum a_n\) is said to be absolutely convergent if the series of absolute values \(\sum |a_n|\) converges. When dealing with absolute convergence, we disregard the sign of individual terms and focus solely on their magnitudes.

A crucial aspect of absolute convergence is its strength; if a series converges absolutely, then it also converges in the usual sense. This property simplifies the analysis of series, as it allows us to first check whether a series converges absolutely. If it does, no further tests for convergence are necessary. Moreover, an absolutely convergent series can be rearranged in any order without affecting its sum, making it much more manageable mathematically.

Conditional Convergence

Contrasting with absolute convergence, conditional convergence occurs when an infinite series \(\sum a_n\) converges, but the series of its absolute values \(\sum |a_n|\) does not. In this scenario, the convergence hinges on the sequence of terms as well as their signs.

Conditionally convergent series are more delicate than their absolutely convergent counterparts. Reordering the terms of a conditionally convergent series can lead to different sums or even make the series diverge. Therefore, understanding whether convergence is absolute or conditional is critical when evaluating the properties and behavior of series.

Ratio Test

The Ratio Test is an invaluable tool for determining the convergence of series, especially when the series involves factorials, exponentials, or other functions that grow quickly. To apply the Ratio Test, we calculate the limit of the ratio of consecutive terms in the absolute values series \(\lim_{n\to\infty} \frac{|a_{n+1}|}{|a_n|}\).

If this limit \(L\) is less than 1, the series converges absolutely. If \(L\) is greater than 1, or the limit does not exist, the series diverges. When the limit \(L\) is exactly 1, the test is inconclusive, and we must resort to other methods to determine convergence. The Ratio Test provides a fast and effective way to assess the convergence of series where the ratio of terms follows a discernable pattern.

Limit of a Sequence

Understanding the concept of the limit of a sequence is fundamental when working with series and the Ratio Test. The limit of a sequence \(\{a_n\}\) as \(n\) approaches infinity, denoted by \(\lim_{n\to\infty} a_n\), represents the value that the sequence's terms get arbitrarily close to as \(n\) increases. In essence, it describes the long-term behavior of the sequence.

If the limit exists and is a finite number, the sequence is said to converge to that number. This concept is crucial when performing the Ratio Test, as we calculate the limit of the ratio of consecutive terms. A clear understanding of limits allows students to accurately evaluate the behavior of series and use the correct tools to determine convergence or divergence.