Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 1: Problem 6

Test the following series for convergence. $$\sum_{n=1}^{\infty} \frac{(-1)^{n} n}{n+5}$$

Short Answer

Expert verified
The series ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline

Step by step solution

01

Identify the Series Type

The given series is ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline
02

Title

The given series is ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Key Concepts

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

Alternating Series
An alternating series is a series where the terms alternate in sign. This means the terms switch from positive to negative, or negative to positive. For example, the series ewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewline ewlineewline is an alternating series. In the given exercise, the series ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline does this because of the factor ewline ewline ewline, which makes the terms flip between positive and negative. Alternating series often need special convergence tests since their behavior can be tricky to analyze.
Convergence Tests
To determine whether an infinite series converges or diverges, we use convergence tests. In the context of alternating series, we often use the Alternating Series Test (AST). The AST states that an alternating series (with terms of the form ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline) converges if the absolute value of the terms satisfies the following properties: ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline Because ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ).
Infinite Series
An infinite series is the sum of the terms of an infinite sequence. This means as we add more terms, the series will infinitely grow or change. For example, ewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlinenewlineewlineewlineewlineewlineewlineewlineewlineewline In the case of our exercise, the series ewlineewline ewline ewline yone term, it will remain infinite. so to determine the convergence, we analyze the behavior of partial sums (the sum of the first N terms) as N grows larger. If the partial sums approach a finite limit, we say the infinite series converges. If not, it diverges.
Mathematical Proofs
A mathematical proof is a logical argument demonstrating that a specific statement is true. Proofs use previously established results and logical reasoning. In analyzing series convergence, proofs often involve using various tests (like the AST or Comparison Test) and logical deductions. ewline ewline ewline ewline ewline ewline ewlineline ewline ewline Because ewline ewline ewline ewline)

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

The following alternating series are divergent (but you are not asked to prove this). Show that \(a_{n} \rightarrow 0 .\) Why doesn't the alternating series test prove (incorrectly) that these series converge? (a) \(\quad 2-\frac{1}{2}+\frac{2}{3}-\frac{1}{4}+\frac{2}{5}-\frac{1}{6}+\frac{2}{7}-\frac{1}{8} \cdots\) (b) \(\quad \frac{1}{\sqrt{2}}-\frac{1}{2}+\frac{1}{\sqrt{3}}-\frac{1}{3}+\frac{1}{\sqrt{4}}-\frac{1}{4}+\frac{1}{\sqrt{5}}-\frac{1}{5} \cdots\)

Use Maclaurin series to do and check your results by computer. $$\lim _{x \rightarrow 0}\left(\frac{1+x}{x}-\frac{1}{\sin x}\right)$$

Find the first few terms of the Maclaurin series for each of the following functions and check your results by computer. $$\cos \left(e^{x}-1\right)$$

Prove that an absolutely convergent series \(\sum_{n=1}^{\infty} a_{n}\) is convergent. Hint: Put \(b_{n}=\) \(a_{n}+\left|a_{n}\right| .\) Then the \(b_{n}\) are nonnegative; we have \(\left|b_{n}\right| \leq 2\left|a_{n}\right|\) and \(a_{n}=b_{n}-\left|a_{n}\right|.\)

Show that \(n ! > 2^{n}\) for all \(n > 3\). Hint: Write out a few terms; then consider what you multiply by to go from, say, \(5 !\) to \(6 !\) and from \(2^{5}\) to \(2^{6}\).
See all solutions

Recommended explanations on Combined Science Textbooks

Synergy

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.

Sign-up for free