/*! 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 47 Derive the geometric series repr... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 47

Derive the geometric series representation of \(1 /(1-x)\) by finding \(a_{0}, a_{1}, a_{2}, \ldots\) such that $$(1-x)\left(a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3}+\cdots\right)=1$$

Short Answer

Expert verified
The series representation of \(\frac{1}{1-x}\) is \(1 + x + x^2 + x^3 + \cdots\).

Step by step solution

01

Understand the problem

We need to find a series such that when multiplied by \((1-x)\), the result is 1. This series will represent \(\frac{1}{1-x}\), a formula often seen in geometric series sum representations.
02

Consider the basic structure

We are given \((1-x)\left(a_{0} + a_{1} x + a_{2} x^2 + a_{3} x^3 + \cdots\right) = 1\). The task is to find the coefficients \(a_0, a_1, a_2, \ldots\) that satisfy this equation.
03

Expand the expression using distribution

Apply the distributive property: \[(1-x)(a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots) = a_0 + (a_1 - a_0)x + (a_2 - a_1)x^2 + (a_3 - a_2)x^3 + \cdots\].
04

Match terms to solve for coefficients

For the equality to hold: - The constant term must be 1: \(a_0 = 1\).- The coefficient of each power of \(x\) must be 0 for all other terms: 1. \(a_1 - a_0 = 0\) gives \(a_1 = a_0 = 1\).2. \(a_2 - a_1 = 0\) gives \(a_2 = a_1 = 1\).3. Continuing this reasoning, \(a_n = 1\) for all \(n\).
05

Write the series representation

The series is: \[a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots = 1 + x + x^2 + x^3 + \cdots\]. This is the geometric series for \(\frac{1}{1-x}\) when \(|x| < 1\).

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.

Series Representation
In mathematics, particularly calculus and analysis, a series representation helps depict a function as an infinite sum of terms. For the function \( \frac{1}{1-x} \), we aim to express it using a series of terms combined together. This representation helps in providing insights into the behavior and approximation of the function over different intervals.
  • A geometric series is one of the simplest forms of series representations, where each term is a constant multiple of the previous one.
  • The task is to express \( \frac{1}{1-x} \) as a sum of power terms in \( x \), specifically \( 1 + x + x^2 + x^3 + \cdots \).
  • This series, we just mentioned, converges entirely when \( |x| < 1 \).
The usefulness of series representation stretches into many areas including approximating functions and solving complex integrals.
Distributive Property
The distributive property is a fundamental arithmetic principle stating that multiplying a number by a sum is the same as doing each multiplication separately. In symbols, it is given as:
\[a imes (b+c) = a imes b + a imes c\]This property allows us to expand expressions systematically, which is crucial in obtaining more complex series or polynomial forms.
  • In the context of our problem, apply the distributive property to \((1-x)\) multiplied by the unknown series \(a_0 + a_1 x + a_2 x^2 + \cdots\).
  • This expansion simplifies understanding by breaking the multiplication into individual terms: \(a_0 + (a_1 - a_0)x + (a_2 - a_1)x^2 + \cdots\).
  • Applying this approach helps to match coefficients and solve for unknowns, such as finding all \(a_n = 1\).
Series Coefficients
The series coefficients \(a_0, a_1, a_2, \ldots\) are the constants that appear in the terms of the series representation. For our problem, they are essential to ensure the equality holds for every power of \(x\).
  • Each coefficient aligns with terms in the equation \((1-x)(a_0 + a_1 x + a_2 x^2 + \cdots) = 1\).
  • Through solving, we find that \(a_0 = 1\), and subsequently all other coefficients like \(a_1 = a_2 = a_3 = \cdots = 1\) resulting from the condition imposed on each power of \(x\).
  • These coefficients are key in shaping the power series to effectively capture the behavior of the function \( \frac{1}{1-x} \).
Power Series Expansion
A power series expansion expresses a function as an infinite sum of terms consisting of powers of a variable. In a sense, it is similar to a polynomial with infinitely many terms.
  • The goal of the expansion is to represent the function \( \frac{1}{1-x} \), which involves expressing it as a sum of powers of \(x\).
  • This power series takes the form \(1 + x + x^2 + x^3 + \cdots\), spanning numerous terms that provide an accurate depiction for \(|x|<1\).
  • Power series are integral in mathematics because they can accurately model behaviors in physics and engineering where polynomial approximations are inadequate.
Understanding the formation and application of a power series aids in diverse fields and even solving differential equations.

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

Find the interval of convergence of each power series. \(\sum_{n=1}^{\infty} \frac{x^{n}}{\sqrt{n}}\)

Suppose that \(\sum a_{n}\) is a conditionally convergent infinite series. For each \(n\), let $$a_{n}^{+}=\frac{a_{n}+\left|a_{n}\right|}{2} \text { and } a_{n}=\frac{a_{n}-\left|a_{n}\right|}{2} \text { . }$$ (a) Explain why \(\Sigma a_{n}^{+}\) consists of the positive terms of \(\Sigma a_{n}\) and why \(\Sigma a_{n}^{-}\) consists of the negative terms of \(\suma_{n}\). (b) Given a real number \(r\), show that some rearrangement of the conditionally convergent series \(\Sigma a_{n}\) converges to \(r\). (Snggestion: If \(r\) is positive, for instance, begin with the first partial sum of the positive series \(\Sigma a_{n}^{+}\) that exceeds \(r\). Then add just enough terms of the negative series \(\Sigma a_{n}\) so that the cumulative sum is less than \(r\). Next add just enough terms of the positive series that the cumulative sum is greater than \(r\), and continuc in this way to define the desired rearrangement.) Why does it follow that this rearranged infinite series converges to \(r\) ?

Determine whether the series in converge absolutely, converge conditionally. or diverge. . \(\sum_{n=1}^{\infty}\left(\frac{\ln n}{n}\right)^{n}\)

Use an infinite series to approximate the indicated number accurate to three decimal places. \(e^{-0.2}\)

Find the interval of convergence of each power series. \(\sum_{n=1}^{\infty} n x^{n}\)
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Pure 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.