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Chapter 11: Problem 15

Determine whether the series is absolutely convergent, conditionally convergent, or divergent. $$ \sum_{n=1}^{\infty}(-1)^{n} \frac{n^{2}+3}{(2 n-5)^{2}} $$

Short Answer

Expert verified
The series is absolutely convergent.

Step by step solution

01

Analyze Terms for Convergence Tests

Identify the general term of the series: \(a_n = (-1)^n \frac{n^2 + 3}{(2n - 5)^2}\). Since the series has \((-1)^n\), it looks like an alternating series. We will need to check for both the absolute and conditional convergence.
02

Absolute Convergence Test

Consider the series of the absolute values: \(|a_n| = \frac{n^2 + 3}{(2n - 5)^2}\). Use the Limit Comparison Test with \(b_n = \frac{1}{n^2}\) since the leading term dominates as \(n\) becomes large. Compute \(\lim_{n \to \infty} \frac{|a_n|}{b_n} = \lim_{n \to \infty} \frac{(n^2 + 3) \cdot n^2}{(2n - 5)^2} = \lim_{n \to \infty} \frac{n^4 + 3n^2}{4n^4 - 20n^3 + 25n^2}\). Both numerator and denominator tend to \(n^4\), leading to \(\lim_{n \to \infty} = \frac{1}{4}\). The series \(\sum 1/n^2\) is convergent, thus by the Limit Comparison Test, \(|a_n|\) converges.
03

Conclusion on Absolute Convergence

Since the series of absolute values \(\sum |a_n|\) converges, the original series is absolutely convergent.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Alternating Series
An alternating series is a sequence of terms that switch signs, typically involving negative and positive terms. This can be seen in our exercise with the given series
  • Notation: \((-1)^{n}\) indicates the alternating nature, which switches the sign of each term as \(n\) increases.
  • Impact: This alternation often influences the series' convergence behavior quite differently than if the signs were all positive.
To determine if an alternating series converges, we often use the
  • Alternating Series Test: If the terms (without their signs) decrease in magnitude to zero as \(n\) approaches infinity, the alternating series converges.
In this exercise, the presence of \((-1)^{n}\) in \(a_n\) alerts us that we're dealing with an alternating series, which is our first clue into exploring its behavior.
Absolute Convergence
Absolute convergence is a stronger form of convergence for a series. If a series converges absolutely, it means that the series formed by taking the absolute values of its terms also converges. This is what we explore when we strip away the \((-1)^{n}\) sign, assessing \(|a_n|\).
  • Why Important: If a series converges absolutely, it guarantees convergence of the series with original alternating signs.
  • Impact on Series: An absolutely convergent series is more stable, meaning changes in order of terms won't affect its sum.
In our solution, we determined that the absolute series, \(|a_n|\), converged based on the Limit Comparison Test, making the original series absolutely convergent. Understanding absolute convergence helps in verifying that a series' behavior remains consistent even when term signs are disregarded.
Limit Comparison Test
The Limit Comparison Test is a powerful tool for analyzing series convergence. It's particularly useful when comparing a series with another whose convergence is already known.
  • Steps: Calculate the limit of the ratio of two series' terms as \(n\) approaches infinity.
  • Criteria: If this limit is finite and non-zero, then both series converge or diverge together.
In our exercise, we performed this test between
  • Our series' absolute terms: \(|a_n| = \frac{n^2 + 3}{(2n - 5)^2}\)
  • Comparison series: \(b_n = \frac{1}{n^2}\)
Since \(\lim_{n \to \infty} \) yielded \(\frac{1}{4}\), and knowing that \(\sum \frac{1}{n^2}\) converges, we established that \(|a_n|\) does too. This method helped simplify our convergence analysis considerably.
Conditional Convergence
Conditional convergence refers to a series that converges, but not absolutely. Simply put, the series converges with its original term signs, but if you were to strip away those signs, the series diverges.
  • Why Matters: Such series are delicate; rearranging them can change their sum due to sensitive balance between positive and negative terms.
  • Contrast: While absolute convergence guarantees convergence regardless of signs, conditional convergence doesn't provide such a safety net.
For the exercise series, we verified it converged absolutely, and thus, it was not just conditionally convergent. However, if it had not been absolutely convergent, exploring conditional convergence would have been key in understanding its behavior.

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