Question

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

Short answer

The series diverges.

Step-by-step solution

Identify the Series Type

The given series is \( \sum_{n=1}^{\infty}(-1)^{n+1}\left(\frac{n-1}{n}\right)^{n} \). This is an alternating series because of the factor \((-1)^{n+1}\).

Apply the Alternating Series Test

For an alternating series \( \sum (-1)^{n+1}b_n \) to converge, we need to check if \( b_n \) is decreasing and if \( \lim_{n \to \infty} b_n = 0 \). Here, \( b_n = \left(\frac{n-1}{n}\right)^n \).

Check Limit of Terms

Evaluate \( \lim_{n \to \infty} \left(\frac{n-1}{n}\right)^n \). Notice that \( \left(1 - \frac{1}{n}\right)^n \approx e^{-1} \) as \( n \to \infty \). Therefore, \( b_n \to e^{-1} eq 0 \).

Conclude on Absolute Convergence

Since \( \lim_{n \to \infty} b_n eq 0 \), the series does not converge absolutely.

Consider Conditional Convergence

For conditional convergence, the alternating series must first converge, which requires \( \lim_{n \to \infty} b_n = 0 \). However, we found that \( b_n \to e^{-1} eq 0 \).

Final Conclusion

Since \( \lim_{n \to \infty} b_n eq 0 \), the alternating series fails to satisfy the requirements for convergence. Hence, this series diverges.

Key concepts

Alternating Series Test

An important method for determining the convergence of a series is the Alternating Series Test. This test is particularly useful for series that exhibit terms alternating in sign, such as \((-1)^{n+1}b_n\). If you have such a series, the key criteria for convergence are:

• The absolute value of the terms, \(b_n\), should be decreasing.
• The limit of the terms as n approaches infinity should be zero, that is, \(\lim_{n\to\infty}b_n = 0\).

Applying the Alternating Series Test can sometimes indicate if a series converges even if its terms do not individually tend to zero. However, if \ \lim_{n\to\infty} b_n \ does not equal zero, the series cannot converge. The test helps us conclude that in such a case, regardless of the alternating nature of the terms, the series diverges.

Absolutely Convergent Series

A series is considered to be absolutely convergent if the series of the absolute values of its terms converges. This means that not only do the series terms alternate in sign, but also, when disregarding their signs, the corresponding positive series converges. For a series \(\sum a_n\), if \ \sum |a_n|\ converges, then we have absolute convergence.
This type of convergence is a strong form, indicating that the series behaves very nicely because the order of terms does not affect convergence. An absolutely convergent series is always convergent, but the reverse is not necessarily true. In the given problem, since the series failed to meet the criteria for term limits. Therefore, it cannot be absolutely convergent.
If absolute convergence is confirmed, then the series automatically converges without suffering from any alternation-related issues. This concept is pivotal in handling complex convergence scenarios.

Conditionally Convergent Series

For a series to be conditionally convergent, it should converge purely because of its alternating sign characteristic rather than the nature of the magnitude of its terms. In simpler words, a series \(\sum a_n\) is conditionally convergent if it converges, but \ \sum |a_n|\ does not.
Let's break it down: First, this type of series must pass the alternating series test, indicating that \(\lim_{n \to \infty} b_n = 0\) and \(b_n\) is decreasing. If these conditions are satisfied, but the absolute series \(\sum |a_n|\) diverges, then we deal with conditional convergence.
In the exercise given, even the condition for ordinary convergence was not fulfilled because \(\lim_{n \to \infty} b_n\) did not approach zero. This ensures no prospect of conditional convergence in this series. It's a clear indicator of how conditional and absolute convergence assessments supplement each other in series analysis.