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Chapter 9: Problem 80

Determine whether the series converges. $$\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{\sqrt{3 n-1}}$$

Short Answer

Expert verified
The series converges by the Alternating Series Test.

Step by step solution

01

Identify the Series Type

The given series is \( \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{\sqrt{3n-1}} \). This is an alternating series because the terms switch between positive and negative due to the \((-1)^{n-1}\) factor.
02

Use the Alternating Series Test

To use the Alternating Series Test, we need to check two conditions for \( a_n = \frac{1}{\sqrt{3n-1}}\): 1. \( a_n \) is positive, which it is since \( \sqrt{3n-1} > 0 \) for every \( n > 0 \).2. \( a_n \) is decreasing, meaning \( a_{n+1} \leq a_n \). Check: \[ \frac{1}{\sqrt{3(n+1)-1}} \leq \frac{1}{\sqrt{3n-1}} \] This simplifies to: \[ \sqrt{3n-1} \leq \sqrt{3n+2} \] which is always true.3. \( \lim_{n \to \infty} a_n = 0 \). Calculate: \( \lim_{n \to \infty} \frac{1}{\sqrt{3n-1}} = 0 \). Since both conditions are satisfied, the series converges by the Alternating Series Test.

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Key Concepts

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

Series Convergence
In mathematics, determining whether a series converges is crucial for understanding the behavior of an infinite sum of terms. A series converges if the sum of its terms approaches a specific value as more and more terms are added. This target value is known as the series' limit.
  • A convergent series implies stability, meaning that the sum does not grow indefinitely or oscillate wildly.
  • Mathematicians use various tests to determine convergence, including the Alternating Series Test, which focuses on series whose terms alternate in sign.
  • Identifying convergence helps predict the behavior of the series, whether it reaches a specific value or diverges to infinity.
Understanding series convergence is vital, as it is applied in real-world problems such as calculating probabilities and solving differential equations.
Alternating Series
An alternating series is a type of series where the signs of the terms alternate between positive and negative. This alternation is generally introduced by multiplication with \( -1^n \) or \( -1^{n-1} \). This peculiar feature can significantly affect the series' convergence.
  • Alternating series often converge more easily or quickly than series with only positive terms.
  • They create a balancing effect, where each subsequent term counteracts the accumulation from previous terms.
  • A widely used method for testing their convergence is the Alternating Series Test, which checks if the term set aside from its sign forms a decreasing sequence and that the limit of these terms as \( n o \) infinity is zero.
Alternating series are important in approximating functions and handling calculations involving oscillating scenarios.
Limit of a Sequence
The limit of a sequence concerns the behavior of the sequence as the number of terms tends toward infinity. Understanding this concept is vital in analyzing series and functions.
  • The limit describes what value, if any, the terms in a sequence approach as the index number grows indefinitely large.
  • If the terms of a sequence approach a specific finite value, the sequence is considered to have a finite limit.
  • Determining this limit is essential in confirming the convergence of an alternating series, as the alternating series test requires the limit of the sequence's terms to be zero.
Grasping this concept is crucial for predicting and verifying long-term behaviors in various mathematical contexts, including calculus and analysis.
Decreasing Sequence
A decreasing sequence is a sequence where each subsequent term is less than or equal to its predecessor. This concept is fundamental for applying the Alternating Series Test.
  • To identify a decreasing sequence, for each term \( a_n\), the condition \( a_{n+1} \leq a_n \) must hold true for every \( n\).
  • The decreasing nature of the terms ensures that each positive term contributes less toward the cumulative sum than the term before it.
  • When presented with an alternating series, it is imperative to verify the decreasing nature of the terms stripped of their signs for convergence via the Alternating Series Test.
A deep understanding of decreasing sequences aids in analyzing and proving the convergence properties of alternating series and other similar problems.

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Most popular questions from this chapter

True or false. Give an explanation for your answer. A power series always converges at at least one point.

Decide if the statements are true or false. Give an explanation for your answer. If all terms \(s_{n}\) of a sequence are less than a million, then the sequence is bounded.

Give an example of: A series \(\sum a_{n}\) such that $$\lim _{n \rightarrow \infty} \frac{\left|a_{n+1}\right|}{\left|a_{n}\right|}=3$$

For what values of \(a\) does the series converge? $$\sum_{n=1}^{\infty} \frac{\ln n}{n^{a}}$$

Which of the sequences I-IV is monotone and bounded for \(n \geq 1 ?\) I. \(s_{n}=10-\frac{1}{n}\) II. \(s_{n}=\frac{10 n+1}{n}\) III. \(s_{n}=\cos n\) IV. \(s_{n}=\ln n\) (a) I (b) I and II (c) II and IV (d) I, II, and III
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