Problem 23 - Essential Calculus Early Transcendentals

Question

If \(\Sigma_{n=0}^{\infty} c_{n} 4^{n}\) is convergent, does it follow that the following series are convergent? $$\text { (a) }\sum_{n=0}^{*} c_{n}(-2)^{n} \quad \text { (b) } \sum_{n=0}^{\infty} c_{n}(-4)^{n}$$

Step-by-Step Solution

Answer

(a) Convergent; (b) Convergent (conditionally due to alternation).

Step 1: Understand the given condition

The series \( \Sigma_{n=0}^{\infty} c_{n} 4^{n} \) is given to be convergent. This implies that the terms \(c_{n} 4^{n}\) must satisfy the condition for convergence, which usually involves the terms decreasing to zero quickly enough.

Step 2: Analyze series (a)

In the series \( \sum_{n=0}^{\infty} c_{n}(-2)^{n} \), we can compare the terms to the original series by noting that \((-2)^n = 4^{n/2}\). Thus, \(c_{n}(-2)^n = c_{n}4^{n/2}\). Since \( 4^{n/2} \), grows slower than \(4^n\), the convergence of the original series implies that \(c_{n}(-2)^{n} \) must approach zero. Therefore, since a convergent series with terms that tend towards zero implies the new series is also convergent.

Step 3: Analyze series (b)

For the series \(\sum_{n=0}^{\infty} c_{n}(-4)^{n}\), note that the terms are \(c_{n}(-4)^n\), which can be rewritten as \(c_{n}(-1)^n \cdot 4^n\). This term simplifies to \((-1)^n\cdot c_n 4^n\). This series resembles the original series except for the factor of \((-1)^n\), which does not affect absolute convergence. As the original series converges absolutely, the introduction of \((-1)^n\) leads to conditional convergence due to alternating series nature.

Key Concepts

Alternating Series

An alternating series is a series where the signs of the terms alternate between positive and negative. This alternation happens because of a factor like

• has a form of \( (-1)^n \), meaning each term switches sign compared to the previous term.
• Alternating series are significant because they can affect convergence, sometimes converging even if their non-alternating counterparts do not.

Understanding can be described with Leibniz's test:

• If the absolute values of the terms decrease monotonically (i.e., consistently) to zero, the series converges.
• This test applies only to series with alternating signs. The current problem involves an alternating series with \((-1)^n\), affecting the series' convergence property.

Thus, for series (b), the alternating nature was added with this alteration, leading to potential conditional rather than absolute convergence.

Absolute Convergence

Absolute convergence in a series is when the series remains convergent even if all terms are replaced with their absolute values. This means:

• A series \(\sum a_n\) converges absolutely if \(\sum |a_n|\) also converges.
• It's a "stronger" form of convergence than conditional convergence.

Understanding absolute convergence helps us analyze series regardless of any alterations to the terms, such as changing their signs. In the given exercise:

• Series \(\sum_{n=0}^{\infty} c_{n}(-4)^{n}\) involves changing the signs of terms.
• However, because the replaced terms yield an absolute convergence scenario in \(\sum_{n=0}^{\infty} c_{n} 4^{n}\), there is an implication that the original series terms satisfy the criteria beyond just absolute magnitude reductions.

This powerful concept limens conditional vs. absolute convergence, making it essential in evaluating intricate series.

Conditional Convergence

Conditional convergence occurs when a series converges, yet it does not converge absolutely. This might sound complex at first, but consider these key points:

• For a series \(\sum a_n\) to be conditionally convergent, \(\sum a_n\) itself converges.
• However, replacing all terms with their absolute values \(\sum |a_n|\) does not converge.

In the context of the given problem, series involving terms with alternate signs easily depict conditional convergence:

• For example, \((-1)^n\) creates a shift that alters convergence, seen in series \(\sum_{n=0}^{\infty} c_{n}(-4)^{n}\).
• Although the original series converges absolutely, introducing alternation changes the scenario to conditional convergence, given only the alternating part.

Thus, understanding conditional convergence is critical in distinguishing cases where a series converges primarily due to its alternating nature.