Problem 23 - Essential Calculus Early Transcendentals

Question

\(19-40=\) Determine whether the series is absolutely convergent, conditionally convergent, or divergent. $$ \sum_{n=0}^{\infty} \frac{(-1)^{n}}{5 n+1} $$

Step-by-Step Solution

Answer

The series is conditionally convergent.

Step 1: Understanding the Series

The given series is \( \sum_{n=0}^{\infty} \frac{(-1)^{n}}{5n+1} \). This is an alternating series since it involves \((-1)^n\). We will analyze it for convergence or divergence using appropriate tests.

Step 2: Applying Alternating Series Test

To apply the Alternating Series Test, we need to check if the terms \( b_n = \frac{1}{5n+1} \) are positive, decreasing, and approach zero as \( n \to \infty \). Clearly, \( b_n > 0 \). Let's check if \( b_n \to 0 \) and if it decreases.

Step 3: Checking \( b_n \to 0 \)

As \( n \to \infty \), the denominator \( 5n + 1 \to \infty \), thus \( b_n = \frac{1}{5n+1} \to 0 \). This condition is satisfied.

Step 4: Checking if \( b_n \) is Decreasing

We need to verify if \( b_{n+1} < b_n \), that is, \( \frac{1}{5n+6} < \frac{1}{5n+1} \). Since \( 5n+6 > 5n+1 \), \( \frac{1}{5n+6} < \frac{1}{5n+1} \), satisfying the condition.

Step 5: Concluding Absolute or Conditional Convergence

Since the series satisfies all the conditions of the Alternating Series Test, it is convergent. To check for absolute convergence, consider \( \sum_{n=0}^{\infty} \left| \frac{(-1)^{n}}{5n+1} \right| = \sum_{n=0}^{\infty} \frac{1}{5n+1} \). Since \( \frac{1}{5n+1} \) does not form a convergent p-series (it behaves like \( \frac{1}{n} \)), the series is not absolutely convergent.

Key Concepts

Series Convergence

In mathematics, the concept of series convergence is crucial when analyzing infinite series. A series can be understood as the sum of the terms of a sequence. In simple terms, if the sum of the infinite terms in a series approaches a specific finite number, we say that the series converges. When dealing with convergence, one common tool is the Alternating Series Test. This test focuses on series where the terms switch signs, such as the series \( \sum_{n=0}^{\infty} \frac{(-1)^{n}}{5n+1} \) from the exercise. Using the Alternating Series Test involves checking two main things:

• The individual terms of the series must tend towards zero as \( n \to \infty \).
• The absolute values of the terms must be decreasing.

If both these conditions are met, the alternating series is convergent. However, this does not imply absolute convergence, which requires further analysis.

Absolute Convergence

Absolute convergence involves a stricter condition compared to simple convergence. A series is absolutely convergent if the series of absolute values is convergent. That is, for the series \( \sum_{n=0}^{\infty} a_n \), if the series \( \sum_{n=0}^{\infty} |a_n| \) also converges, then the original series is absolutely convergent. This property is beneficial because when a series is absolutely convergent, it converges even if the order of its terms is rearranged—a feature not shared by conditionally convergent series.In the context of the given series \( \sum_{n=0}^{\infty} \frac{(-1)^{n}}{5n+1} \), we look at its absolute version: \( \sum_{n=0}^{\infty} \frac{1}{5n+1} \). By comparing \( \frac{1}{5n+1} \) to the harmonic series \( \frac{1}{n} \), which is divergent, we conclude that this series is not absolutely convergent since its behavior is similar to that of the divergent harmonic series.

Conditional Convergence

Conditional convergence of a series occurs when a series fulfills the criteria for convergence, but fails the criteria for absolute convergence. This means that the series converges when taken as a whole, but the series of its absolute values diverges.The series \( \sum_{n=0}^{\infty} \frac{(-1)^{n}}{5n+1} \) serves as a perfect example. We have determined that it converges by applying the Alternating Series Test successfully, because:

• The terms \( \frac{1}{5n+1} \) go to zero as \( n \to \infty \).
• The terms are decreasing in absolute value.

However, in evaluating absolute convergence, we found that \( \sum_{n=0}^{\infty} \frac{1}{5n+1} \) behaves like a harmonic series, which diverges. This makes the original series conditionally convergent—convergent as it stands, but not robust to the removal of the alternating signs.