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 and does not converge absolutely or conditionally.

Step-by-step solution

Analyze General Term Behavior

First, consider the general term of the series, which is \((-1)^{n+1} \left(\frac{n+1}{n}\right)^{n}\). We must check the behavior of its absolute value for convergence tests. The absolute value of the general term is \(\left(\frac{n+1}{n}\right)^{n}\).

Apply Limit Test for Exponential Expression

Examine \( \lim_{n \to \infty} \left(\frac{n+1}{n}\right)^{n} \). Simplify this expression using known limits and properties of exponents: \(\left(1 + \frac{1}{n}\right)^{n} \to e\) as \(n \to \infty\). Therefore, \(\left(\frac{n+1}{n}\right)^{n}\) behaves similarly and approaches \(e\) as \(n\) goes to infinity.

Determine Absolute Convergence

Since \(\lim_{n \to \infty} \left(\frac{n+1}{n}\right)^{n} = e\), and \(e > 1\), the absolute value of the general term does not tend to zero. This implies the series \(\sum_{n=1}^{\infty} \left(\frac{n+1}{n}\right)^{n}\) diverges. Hence, the original series does not converge absolutely.

Check Conditional Convergence

To determine conditional convergence, we now consider the series with the original sign pattern: \(\sum_{n=1}^{\infty} (-1)^{n+1} \left(\frac{n+1}{n}\right)^{n}\). An alternating series \(\sum (-1)^n a_n\) converges if \(a_n\) decreasingly approaches zero. Here, however, the absolute term \(\left(\frac{n+1}{n}\right)^{n}\) does not approach zero, hence the alternating series does not converge conditionally either.

Key concepts

Absolute Convergence

Absolute convergence occurs when the series of absolute values converge. To check for absolute convergence, replace any negative terms with their positive equivalents. This process helps in deciding if a series will converge regardless of any sign changes.

For example, consider the series:

• The absolute value of the general term is \left(\frac{n+1}{n}\right)^{n}\.
• Even though the series contains alternating signs due to the term \((-1)^{n+1}\), we focus on \left(\frac{n+1}{n}\right)^{n}\.

To check for absolute convergence, examine if \( \sum_{n=1}^{\infty} \left(\frac{n+1}{n}\right)^{n} \) converges.
The solution shows that the limit of \( \left(\frac{n+1}{n}\right)^{n} \) as \( n \) approaches infinity is \( e \), which is greater than 1. Thus, the absolute value of the terms does not approach zero, and the series does not converge absolutely.

Conditional Convergence

Conditional convergence occurs when a series converges only because of the alternation of signs, even though its absolute value diverges. This usually applies to alternating series.

For our series,

• The criterion for conditional convergence, such as the Alternating Series Test, must check that the absolute values of the terms approach zero and the sequence is decreasing.

Unfortunately, in our case, while the signs alternate, the expression \( \left(\frac{n+1}{n}\right)^{n} \) doesn't tend to zero as \( n \) increases.
Since the terms do not decrease to zero, the original series \( \sum_{n=1}^{\infty} (-1)^{n+1} \left(\frac{n+1}{n}\right)^{n} \) does not converge conditionally either.

Alternating Series Test

The Alternating Series Test (AST) is a useful test to determine the convergence of series with terms that alternate in sign. This test checks two main criteria:

• The absolute value of the terms in the sequence must be decreasing.
• As \( n \) approaches infinity, the limit of the absolute value of the term \( a_n \) must be zero.

In the series given, these conditions are vital to examining potential convergence. Despite the series having an alternating pattern, we find that \( \left(\frac{n+1}{n}\right)^{n} \) doesn't decrease or approach zero.
This failure to meet the criteria results in the series not converging by the AST. This solidifies the conclusion that the series neither converges absolutely nor conditionally.