Question

An alternating series \(\sum_{n=i}^{\infty} a_{n}\) is given. (a) Determine if the series converges or diverges. (b) Determine if \(\sum_{n=0}^{\infty}\left|a_{n}\right|\) converges or diverges. (c) If \(\sum_{n=0}^{\infty} a_{n}\) converges, determine if the convergence is conditional or absolute. $$\sum_{n=0}^{\infty}(-1)^{n} \frac{n+5}{3 n-5}$$

Short answer

The series diverges; the absolute series diverges as well.

Step-by-step solution

Examine the Series for Convergence (Alternating Series Test)

The given series is \( \sum_{n=0}^{\infty} (-1)^n \frac{n+5}{3n-5} \). We need to apply the Alternating Series Test (AST) to check its convergence. The AST requires that \( b_n = \frac{n+5}{3n-5} \) is positive, decreasing, and \( \lim_{n \to \infty} b_n = 0 \).1. **Positivity:** For \( n \geq 2 \), \( b_n \) is positive since \( 3n-5 \) becomes positive.2. **Decreasing:** Check if \( b_{n+1} < b_n \): \[ \frac{(n+1)+5}{3(n+1)-5} < \frac{n+5}{3n-5} \]. Cross-multiply and simplify to confirm this holds.3. **Limit:** Compute \( \lim_{n \to \infty} b_n = \lim_{n \to \infty} \frac{n+5}{3n-5} = \frac{1}{3} eq 0 \).Since the limit is not zero, the series does not satisfy the necessary condition for convergence by the AST, thus it diverges.

Check Convergence of Absolute Series

Now, consider the absolute series, \( \sum_{n=0}^{\infty} \left|(-1)^n \frac{n+5}{3n-5} \right| = \sum_{n=0}^{\infty} \frac{n+5}{3n-5} \). The term \( \frac{n+5}{3n-5} \) does not tend to zero as shown before (it tends to \( \frac{1}{3} \)), so the series diverges by the Divergence Test (since the terms do not approach zero).

Determining Conditional or Absolute Convergence

From Step 1, we found that the alternating series diverges. Since the series does not converge, we cannot discuss conditional or absolute convergence. Thus, it diverges without possibility of convergence.

Key concepts

Alternating Series Test

An alternating series is a series whose terms alternate between positive and negative. The Alternating Series Test (AST) is a handy tool used to determine if such a series converges. For the AST to be applicable, three conditions must be met:

• The terms of the series, ignoring their signs, must be positive: This means each term in the sequence is greater than zero after ignoring the alternating factor like \((-1)^n\).
• The terms must be decreasing: This involves checking if \(b_{n+1} < b_n\). This can be verified by simplifying the respective inequality.
• The limit of the terms must approach zero as \(n\) approaches infinity: Mathematically, \(\lim_{n \to \infty} b_n = 0\).

When all these conditions are satisfied, the series converges. In the example series given, although the first two conditions meet, the limit condition is not passing because the limit tends to \(\frac{1}{3}\), not zero. Therefore, the given series does not converge according to the AST.

conditional convergence

Conditional convergence occurs when a series converges but does not converge absolutely. In simpler terms, the series converges if we consider the alternating signs, yet it would diverge if all terms were made positive.
For a series to show conditional convergence, it must meet the AST requirements but fail to do so under absolute terms.
Let's consider our series:

• The series \(\sum (-1)^n \frac{n+5}{3n-5}\) fails to converge by AST, hence cannot showcase conditional convergence.
• If a series had converged by AST, you would then analyze the absolute series \(\sum |a_n|\) to check if it diverges to affirm conditional convergence.

Since in this situation the alternating series itself diverges, there is no conditional convergence to assess.

absolute convergence

Absolute convergence is a stronger form of convergence. If a series absolutely converges, it implies that the series also converges conditionally.
To determine absolute convergence, evaluate the series comprising the absolute values of its terms: \(\sum |a_n|\).

• If this "absolute" series converges, the original series converges absolutely.
• If the "absolute" series diverges but the original converges, we can only speak of conditional convergence, which is not the case here.

In our example:

The series \(\sum |(-1)^n \frac{n+5}{3n-5}| = \sum \frac{n+5}{3n-5}\) does not approach zero, leading to divergence by the Divergence Test.

Therefore, absolute convergence is not achieved in this scenario.

Divergence Test

When dealing with infinite series, an essential starting point is the Divergence Test. This test checks if the terms of the series itself approach zero as \(n\) approaches infinity.
The Divergence Test states:

• If \(\lim_{n \to \infty} a_n \eq 0\), or the limit does not exist, the series must diverge.

It's important to remember the Divergence Test cannot confirm convergence but can easily identify divergence.
In the provided exercise, the absolute series was examined using the Divergence Test. As the series terms tend to \(\frac{1}{3}\) instead of zero, divergence is clear. Thus, the Divergence Test confirms that the absolute series does not converge, reinforcing the conclusion that the entire alternating series diverges.