/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 35 Which of the series in Exercises... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 35

Which of the series in Exercises \(15 - 48\) converge absolutely, which converge, and which diverge? Give reasons for your answers. $$ \sum _ { n = 1 } ^ { \infty } \frac { \cos n \pi } { n \sqrt { n } } $$

Short Answer

Expert verified
The series converges absolutely because it is an alternating series with terms \( \left| a_n \right| = \frac{1}{n^{3/2}} \), which is convergent.

Step by step solution

01

Analyze the General Term

The general term of the series is given by \( a_n = \frac{\cos(n\pi)}{n\sqrt{n}} \). Note that \( \cos(n\pi) \) alternates between \( -1 \) and \( 1 \) depending on whether \( n \) is even or odd. Hence, the series is alternating.
02

Check Absolute Convergence

We check absolute convergence by considering \( \left| a_n \right| = \frac{1}{n^{3/2}} \). The series \( \sum_{n=1}^{\infty} \frac{1}{n^{3/2}} \) is a p-series with \( p = 3/2 > 1 \). Therefore, \( \sum_{n=1}^{\infty} \left| a_n \right| \) converges.
03

Conclusion on Absolute Convergence

Since the series \( \sum_{n=1}^{\infty} \left| a_n \right| \) converges, the original series \( \sum_{n=1}^{\infty} a_n \) converges absolutely.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Alternating Series
An alternating series is a series whose terms alternate in sign. In other words, the sequence of terms goes back and forth between positive and negative values. This can happen when the terms of the series are governed by a factor such as \( \cos(n\pi) \) which equals \(-1\) or \(1\) depending on whether \(n\) is odd or even, respectively. Alternating series are important because they can sometimes converge even when the corresponding series of absolute values does not.
  • To analyze an alternating series, one must consider both convergence (whether it approaches a finite limit) and absolute convergence (whether the sum of absolute values converges).
  • For example, the Leibniz Test is a common method to test the convergence of alternating series.
Altering series are frequently studied due to these interesting properties, which help understand deeper convergence behaviors of series.
P-Series
A p-series is a specific type of mathematical series characterized by \[ \sum_{n=1}^{\infty} \frac{1}{n^p} \]where \( p \) is a positive constant. The behavior of a p-series depends heavily on the value of \( p \).
  • If \( p > 1 \), the p-series converges.
  • If \( p \leq 1 \), the series diverges.
This series is widely used as a benchmark for analyzing other series due to its simple form and well-defined convergence criteria. For example, with \( p=3/2 \), as in our exercise, the series converges. This allows us to make conclusions about the convergence of more complex series when they can be compared to a p-series.
Convergence Tests
Convergence tests are techniques used to determine whether a series converges. There are multiple tests available, each suited for different types of series. Some of the common ones are:
  • Alternating Series Test (Leibniz Test): This checks the convergence of alternating series based on the size of terms and whether they decrease to zero.
  • Ratio Test: Frequently used to assess series with factorials or exponential functions.
  • Root Test: Useful for series involving roots or exponential growth.
  • P-Series Test: Specifically for p-series, based on the power of the denominator.
For each series, picking the correct test depends on the form and characteristics of the series' general term. For example, in our exercise, the p-series test confirmed convergence by comparing to a known convergent p-series.
Series Analysis
Series analysis involves a thorough evaluation of series to determine their convergence, divergence, or absolute convergence properties. This can include identifying and applying suitable convergence tests, recognizing underlying patterns, or decomposing a series into simpler components for analysis.
Key steps in series analysis might include:
  • Identifying whether the series is alternating or not.
  • Checking for absolute convergence by examining a related p-series or using integral tests.
  • Concluding based on the results of convergence tests, making sure to justify the use of each method employed.
By systematically applying these techniques, one gains a comprehensive understanding of a series' behavior.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Which of the series in Exercises 13 46 converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series' convergence or divergence.) $$ \sum_{n=1}^{\infty} \frac{e^{n}}{10+e^{n}} $$

\begin{equation} \begin{array}{l}{\text { a. Use Taylor's formula with } n=2 \text { to find the quadratic }} \\ {\text { approximation of } f(x)=(1+x)^{k} \text { at } x=0(k \text { a constant })} \\ {\text { b. If } k=3, \text { for approximately what values of } x \text { in the interval }} \\ {[0,1] \text { will the error in the quadratic approximation be less }} \\ {\text { than } 1 / 100 ?}\end{array} \end{equation}

Which of the sequences converge, and which diverge? Give reasons for your answers. $$ a_{n}=\frac{2^{n}-1}{3^{n}} $$

The sum of the series \(\sum_{n=0}^{\infty}\left(n^{2} / 2^{n}\right)\) To find the sum of this series, express 1\(/(1-x)\) as a geometric series, differentiate both sides of the resulting equation with respect to \(x,\) multiply both sides of the result by \(x\) , differentiate again, multiply by \(x\) again, and set \(x\) equal to 1\(/ 2 .\) What do you get?

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} \begin{equation} \begin{array}{l}{\text { to generate the first four nonzero terms of the Taylor series }} \\ {\text { for sin }^{-1} x . \text { What is the radius of convergence? }}\end{array} \end{equation} b. Series for \(\cos ^{-1} x\) Use your result in part (a) to find the first five nonzero terms of the Taylor series for \(\cos ^{-1} x .\)
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.