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Chapter 10: Problem 2

Determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{1}{n^{3 / 2}} $$

Short Answer

Expert verified
The series converges by the Alternating Series Test.

Step by step solution

01

Identify the Alternating Series

The series given is \( \sum_{n=1}^{\infty}(-1)^{n+1} \frac{1}{n^{3 / 2}} \). This series alternates because of the factor \((-1)^{n+1}\).
02

Apply the Alternating Series Test

The Alternating Series Test requires that the absolute value of the terms \(b_n = \frac{1}{n^{3 / 2}}\) must be positive, decreasing, and approach 0 as \(n\) approaches infinity.
03

Check Positivity

The terms \(b_n = \frac{1}{n^{3 / 2}}\) are positive for all \(n \geq 1\).
04

Check Decreasing Nature

To verify \(b_n\) is decreasing, we check if \( \frac{1}{n^{3 / 2}} > \frac{1}{(n+1)^{3 / 2}} \) for all \(n\). This inequality holds because as \(n\) increases, \(n^{3/2}\) increases, making \(b_n\) decrease.
05

Calculate Limit

Check if \( \lim_{n \to \infty} b_n = 0 \). The limit \( \lim_{n \to \infty} \frac{1}{n^{3/2}} = 0 \) as the denominator grows indefinitely.
06

Conclusion from the Test

Since \(b_n\) is positive, decreasing, and approaches 0, the series satisfies the conditions of the Alternating Series Test, therefore, it converges.

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Key Concepts

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

Convergence
Convergence in series analyzes the behavior of an infinite series as more and more terms are added. A series converges if the sum of its terms approaches a finite number as the number of terms goes to infinity. This is an essential concept because it helps us understand whether an infinite series has a definitive sum or not.
  • The concept of limit is crucial in determining convergence. If the limit of the series as the number of terms increases is finite, the series converges.
  • Various tests can help determine convergence. These include the Alternating Series Test, Ratio Test, and more.
  • Convergence is important in calculus since it deals with sums that extend indefinitely, ensuring that they make sense mathematically.
Understanding convergence allows us to manipulate and work with series confidently, knowing whether they will behave in a predictable manner or not.
Series
A series is essentially the sum of the terms of a sequence. When we talk about a series in mathematics, we are usually referring to an expression that involves adding up infinitely many terms from a sequence.
  • Series can be finite or infinite. A finite series has a limited number of terms, whereas an infinite series continues indefinitely.
  • To study an infinite series, we often look at its behavior as the number of terms grows, checking whether it converges to a specific limit.
  • Types of series include arithmetic, geometric, and alternating series, each with unique properties and methods for analysis.
The mathematical exploration of series helps in various scientific and engineering fields where continuous sums model real-world phenomena.
Decreasing Sequence
A decreasing sequence is a sequence where each term is less than or equal to the one before it. In the context of the Alternating Series Test, recognizing a decreasing sequence is fundamental.
  • For a sequence to be decreasing, for every consecutive pair of terms, the subsequent term must not be larger than the preceding term.
  • This property is important in alternating series because one of the test's conditions requires that the sequence of absolute values of terms, without the alternating sign, must decrease.
  • A simple way to check for decreasing nature is by comparing terms, ensuring that extra terms lead to smaller values.
A decreasing sequence often indicates that the series' terms are approaching a stable point, aiding in the determination of convergence or divergence through tests like the Alternating Series Test.

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Most popular questions from this chapter

a. Use the binomial series and the fact that \begin{equation}\frac{d}{d x} \sin ^{-1} x=\left(1-x^{2}\right)^{-1 / 2}\end{equation} to generate the first four nonzero terms of the Taylor series for \(\sin ^{-1} x .\) What is the radius of convergence? b. Series for \(\cos ^{-1} x\) Use your result in part (a) to find the first five nonzero terms of the Taylor series for \(\cos ^{-1} x .\)

Obtain the Taylor series for 1\(/(1+x)^{2}\) from the series for \(-1 /(1+x) .\)

Which of the series converge, and which diverge? Use any method, and give reasons for your answers. \begin{equation}\sum_{n=1}^{\infty} \frac{1}{n \sqrt[n]{n}}\end{equation}

Computer Explorations Taylor's formula with \(n=1\) and \(a=0\) gives the linearization of a function at \(x=0 .\) With \(n=2\) and \(n=3\) we obtain the standard quadratic and cubic approximations. In these exercises we explore the errors associated with these approximations. We seek answers to two questions: \begin{equation} \begin{array}{l}{\text { a. For what values of } x \text { can the function be replaced by each }} \\ {\text { approximation with an error less than } 10^{-2} \text { ? }} \\ {\text { b. What is the maximum error we could expect if we replace the }} \\ {\text { function by each approximation over the specified interval? }}\end{array} \end{equation} Using a CAS, perform the following steps to aid in answering questions (a) and (b) for the functions and intervals in Exercises \(53-58 .\) \begin{equation} \begin{array}{l}{\text { Step } 1 : \text { Plot the function over the specified interval. }} \\ {\text { Step } 2 : \text { Find the Taylor polynomials } P_{1}(x), P_{2}(x), \text { and } P_{3}(x) \text { at }} \\\ {x=0 .}\\\\{\text { Step } 3 : \text { Calculate the }(n+1) \text { st derivative } f^{(n+1)}(c) \text { associated }} \\ {\text { with the remainder term for each Taylor polynomial. }} \\ {\text { Plot the derivative as a function of } c \text { over the specified interval }} \\ {\text { and estimate its maximum absolute value, } M .}\\\\{\text { Step } 4 : \text { Calculate the remainder } R_{n}(x) \text { for each polynomial. }} \\ {\text { Using the estimate } M \text { from Step } 3 \text { in place of } f^{(n+1)}(c), \text { plot }} \\ {R_{n}(x) \text { over the specified interval. Then estimate the values of }} \\ {x \text { that answer question (a). }}\\\\{\text { Step } 5 : \text { Compare your estimated error with the actual error }} \\ {E_{n}(x)=\left|f(x)-P_{n}(x)\right| \text { by plotting } E_{n}(x) \text { over the specified }} \\ {\text { interval. This will help answer question (b). }} \\ {\text { Step } 6 : \text { Graph the function and its three Taylor approximations }} \\ {\text { together. Discuss the graphs in relation to the information }} \\ {\text { discovered in Steps } 4 \text { and } 5 .}\end{array} \end{equation} $$f(x)=e^{-x} \cos 2 x, \quad|x| \leq 1$$

Use series to evaluate the limits. \begin{equation} \lim _{x \rightarrow 0} \frac{\ln \left(1+x^{3}\right)}{x \cdot \sin x^{2}} \end{equation}
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