/*! 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 19 Does the series converge or dive... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 19

Does the series converge or diverge? $$\sum_{n=1}^{\infty} \frac{3}{(2 n-1)^{2}}$$

Short Answer

Expert verified
The series converges because it's a p-series with p > 1.

Step by step solution

01

Identify the Series Type

The given series is \( \sum_{n=1}^{\infty} \frac{3}{(2n-1)^2} \). Notice that it resembles a p-series because it is in the form \( \sum \frac{1}{n^p} \). Here, the general term is \( \frac{3}{(2n-1)^2} \).
02

Rewriting the Series

Rewrite the series to closely examine its behavior by factoring out the constant:\[\sum_{n=1}^{\infty} \frac{3}{(2n-1)^2} = 3 \sum_{n=1}^{\infty} \frac{1}{(2n-1)^2}.\]
03

Recognize the P-series

The series \( \sum_{n=1}^{\infty} \frac{1}{(2n-1)^2} \) is a p-series, where the general form is \( \sum \frac{1}{k^p} \) with \( k = 2n - 1 \) and \( p=2 \).
04

Apply the P-series Test

In a p-series \( \sum \frac{1}{k^p} \), it converges if \( p > 1 \). Here \( p = 2 \), which is greater than 1, indicating convergence.
05

Conclusion on Convergence

Since \( \sum_{n=1}^{\infty} \frac{1}{(2n-1)^2} \) converges by the p-series test, the original series \( \sum_{n=1}^{\infty} \frac{3}{(2n-1)^2} \) also converges, because a constant multiple of a convergent series is also convergent.

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.

Convergent Series
A convergent series is a sequence of numbers where the sum approaches a specific finite value as more terms are added. This means that as you keep calculating the sum by adding more terms, the result becomes closer to a particular number. A classic example is the harmonic series, which converges if the sequence's terms decrease rapidly enough. For a series to be labeled as convergent:
  • The sequence of partial sums must approach a finite limit.
  • The terms of the series must get smaller as you progress through the series.
The given series \( \sum_{n=1}^{\infty} \frac{3}{(2n-1)^2} \) is determined to be convergent based on the characteristics derived from the p-series test. This means its sum approaches a specific number as you move towards infinity.
P-series Test
The p-series test is a common and useful tool to determine the convergence or divergence of series in the form \( \sum \frac{1}{n^p} \). It's an essential part of understanding series convergence:
  • A p-series converges when \( p > 1 \).
  • A p-series diverges when \( p \leq 1 \).
For the given series \( \sum_{n=1}^{\infty} \frac{1}{(2n-1)^2} \), this can be compared to a p-series with \( p = 2 \). Since 2 is greater than 1, the series passes the p-series test and converges. The p-series test is favored for its simplicity and effectiveness in determining the behavior of similar series.
Series Convergence
Series convergence is about determining whether adding up all the terms in a series will settle on a specific value or not. A series converges when the infinite addition of terms results in a finite limit. This concept is pivotal in calculus and analysis:
  • To determine convergence, apply tests such as the p-series test, ratio test, or comparison test.
  • Each test has specific conditions and applications for when it is most useful.
In addressing the question \( \sum_{n=1}^{\infty} \frac{3}{(2n-1)^2} \), the series is found to converge using the p-series test. This knowledge helps predict how the sum will behave even if the number of terms becomes very large, which is crucial in real-world applications and mathematical problem solving.

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

Determine whether the series converges. $$\sum_{n=1}^{\infty} \frac{n 2^{n}}{3^{n}}$$

Determine whether the series converges. $$\sum_{n=1}^{\infty} \frac{2 n+1}{\sqrt{3 n^{3}-2}}$$

Determine whether the series converges. $$\sum_{n=1}^{\infty} \frac{1}{n^{2}} \tan \left(\frac{1}{n}\right)$$

Let \(\sum a_{n}\) be a series that converges to a number \(L\) and whose terms are all positive. We show that any rearrangement \(\sum c_{n}\) of this series also converges to \(L\) (a) Show that \(S_{n}K\) (c) Use parts (a) and (b) to show \(\sum c_{n}\) converges to \(L\)

Explain what is wrong with the statement. The sum of the infinite geometric series \(1-\frac{3}{2}+\frac{9}{4}-\) \(\frac{27}{8}+\dots\) is \(\frac{1}{1+3 / 2}=\frac{2}{5}\)
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

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.