/*! 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 70 In Exercises \(57 - 82 ,\) use a... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 70

In Exercises \(57 - 82 ,\) use any method to determine whether the series converges or diverges. Give reasons for your answer. $$ 1 - \frac { 1 } { 8 } + \frac { 1 } { 64 } - \frac { 1 } { 512 } + \frac { 1 } { 4096 } - \cdots $$

Short Answer

Expert verified
The series converges to \( \frac{8}{9} \).

Step by step solution

01

Identify the series

The given series is: \[ 1 - \frac{1}{8} + \frac{1}{64} - \frac{1}{512} + \cdots \]This is an infinite series with alternating terms. Each term can be expressed as |\( a_n = \frac{(-1)^n}{8^n} \)|.
02

Determine the type of series

Notice that each term in the series, except the first, involves the factor \( \frac{1}{8} \). The general term can be written as \( a_n = \left(\frac{-1}{8}\right)^n \) for \( n \geq 0 \). This indicates that the series has a form similar to a geometric series.
03

Find the common ratio

A geometric series can be expressed in the form \( a_1 + a_1 r + a_1 r^2 + \cdots \) with common ratio \( r \). Here, compare consecutive terms \[ \frac{1/8}{1} = \frac{1}{8}, \quad \frac{1/64}{-1/8} = \frac{-1}{8}, \quad \frac{1/512}{1/64} = \frac{-1}{8} \]The common ratio \( r \) is \( -\frac{1}{8} \).
04

Check convergence criteria

A geometric series converges if the absolute value of the common ratio \( |r| < 1 \). Here, \[ \left| -\frac{1}{8} \right| = \frac{1}{8} < 1 \]Therefore, the series converges.
05

Calculate the sum (if convergent)

The sum \( S \) of an infinite convergent geometric series is given by \[ S = \frac{a_1}{1 - r} \]where \( a_1 = 1 \) and \( r = -\frac{1}{8} \). Substituting these values, we find:\[ S = \frac{1}{1 - (-\frac{1}{8})} = \frac{1}{1 + \frac{1}{8}} = \frac{1}{\frac{9}{8}} = \frac{8}{9} \].

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.

Geometric Series
A geometric series is a special type of series where each term after the first is the product of the previous term and a constant called the common ratio. This series has a clear and easily recognizable pattern. It's written in the form:
  • \[ a_1 + a_1 r + a_1 r^2 + a_1 r^3 + \ \]
where \( a_1 \) is the first term, and \( r \) is the common ratio.
Geometric series can either be finite or infinite. For an infinite geometric series to converge, the absolute value of the common ratio \( |r| \) must be less than 1. If this is the case, not only does the series converge, but the sum can also be calculated using the formula:
  • \[ S = \frac{a_1}{1 - r} \]
Understanding the behavior of geometric series is crucial in many fields, including finance, physics, and computer science, where they frequently model growth and decay processes.
Alternating Series
An alternating series is characterized by terms that alternate in sign. This means the sequence of terms switches between positive and negative, like the series:
  • \[ 1, -\frac{1}{8}, \frac{1}{64}, -\frac{1}{512}, \ldots \]
Alternating series often converge even when their non-alternating counterparts do not. A key criterion for the convergence of an alternating series is given by the Alternating Series Test, which states that if the absolute value of the terms decreases steadily to zero, the series converges.
This makes them interesting and useful for approximations and series summation because they can provide insights and estimates in mathematical computations and problem-solving scenarios. Recognizing that a series is alternating is often the first step toward proving its convergence and estimating its sum.
Common Ratio
The common ratio in a geometric series is the factor that each term in the series is multiplied by to obtain the next term. It plays a critical role in determining the behavior of the series:
  • For convergence in an infinite series, \( |r| < 1 \).
  • For divergence, \( |r| \geq 1 \).
In the example series, the calculated common ratio is \(-\frac{1}{8}\), indicating that each term is negative one eighth of the previous term.
Identifying the common ratio allows you to apply the geometric series formula for convergence, and when looking at alternating series, it can also help determine the series' oscillating nature.
Infinite Series
An infinite series is a sum of infinitely many terms. Unlike finite series, which sum a limited number of terms, infinite series continue indefinitely. Key to understanding infinite series is knowing when they converge or diverge.
Full convergence is a desirable property for practical applications because it implies a finite sum. For the series in question, the infinite nature doesn鈥檛 prevent summation, thanks to the convergence from the alternating terms and a common ratio with an absolute value less than 1.
This series demonstrates how infinite series can be tackled using known tests, like the geometric series test for convergence. Without convergence, an infinite series doesn鈥檛 sum up to a meaningful value but diverges instead, meaning it leads to infinities or undefined results.

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

Uniqueness of convergent power series $$ \begin{array}{l}{\text { a. Show that if two power series } \sum_{n=0}^{\infty} a_{n} x^{n} \text { and } \sum_{n=0}^{\infty} b_{n} x^{n}} \\\ {\text { are convergent and equal for all values of } x \text { in an open }} \\ {\text { interval }(-c, c), \text { then } a_{n}=b_{n} \text { for every } n . \text { Hint: Let }} \\ {f(x)=\sum_{n=0}^{\infty} a_{n} x^{n}=\sum_{n=0}^{\infty} b_{n} x^{n} . \text { Differentiate term by term to }} \\ {\text { show that } a_{n} \text { and } b_{n} \text { both equal } f^{(n)}(0) /(n !) . )}\\\\{\text { b. Show that if } \sum_{n=0}^{\infty} a_{n} x^{n}=0 \text { for all } x \text { in an open interval }} \\ {(-c, c), \text { then } a_{n}=0 \text { for every } n .}\end{array} $$

Sequences generated by Newton's method Newton's method, applied to a differentiable function \(f(x),\) begins with a starting value \(x_{0}\) and constructs from it a sequence of numbers \(\left\\{x_{n}\right\\}\) that under favorable circumstances converges to a zero of \(f .\) The recursion formula for the sequence is $$ x_{n+1}=x_{n}-\frac{f\left(x_{n}\right)}{f^{\prime}\left(x_{n}\right)} $$ a. Show that the recursion formula for \(f(x)=x^{2}-a, a>0\) can be written as \(x_{n+1}=\left(x_{n}+a / x_{n}\right) / 2\) b. Starting with \(x_{0}=1\) and \(a=3,\) calculate successive terms of the sequence until the display begins to repeat. What number is being approximated? Explain.

Use the following steps to prove that the binomial series in Equation \((1)\) converges to \((1+x)^{m}\). \begin{equation}\begin{array}{l} \\ {\text { a. Differentiate the series }}\end{array}\end{equation} \begin{equation}\quad \quad \quad f(x)=1+\sum_{k=1}^{\infty} \left( \begin{array}{l}{m} \\ {k}\end{array}\right) x^{k}\end{equation} to show that \begin{equation}f^{\prime}(x)=\frac{m f(x)}{1+x}, \quad-1< x < 1.\end{equation} \begin{equation} \begin{array}{l}{\text { b. Define } g(x)=(1+x)^{-m} f(x) \text { and show that } g^{\prime}(x)=0} \\ {\text { c. From part (b), show that }}\end{array}\end{equation} \begin{equation}f(x)=(1+x)^{m}\end{equation}

Assume that the series \(\sum a_{n} x^{n}\) converges for \(x=4\) and diverges for \(x=7 .\) Answer true \((T),\) false \((F),\) or not enough information given \((\mathrm{N})\) for the following statements about the series. $$ \begin{array}{l}{\text { a. Converges absolutely for } x=-4} \\ {\text { b. Diverges for } x=5} \\ {\text { c. Converges absolutely for } x=-8.5} \\\ {\text { d. Converges absolutely for } x=-8.5} \\ {\text { e. Diverges for } x=8} \\ {\text { f. Diverges absolutely for } x=0} \\ {\text { g. Converges absolutely for } x=-7.1}\end{array} $$

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.) $$ \sum_{n=1}^{\infty} \frac{2}{1+e^{n}} $$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Applied 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.