/*! 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 54 \(R_{n}(x)=\frac{\frac{1}{(1-x)^... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 54

\(R_{n}(x)=\frac{\frac{1}{(1-x)^{(n+1)}(c)}}{(n+1) !} x^{n+1}\)

Short Answer

Expert verified
#tag_title#Answer#tag_content# 1. \((1-x)^{(n+1)}\) - determines the behavior of the function as x approaches 1, causing an infinite limit. 2. \(c\) - serves as a scaling factor determining the steepness of the function. 3. \((n+1)!\) - balances the equation between the rising power of x and the growth of the factorial. 4. \(x^{n+1}\) - represents consecutive powers of x in the Taylor series approximation.

Step by step solution

01

Identify the variables

In this equation, there are three variables: x, n, and c. - x can be any real number - n is a non-negative integer, as indicated by the n+1 factorial - c is a constant Keep in mind that the main variable we will work with is x, whereas n and c are constants.
02

Understand the components of the function

To better understand and work with this function, let's identify its components: 1. \((1-x)^{(n+1)}\) - The denominator of the term inside the larger fraction 2. \(c\) - The constant multiplied in the denominator of the fraction inside the larger fraction 3. \((n+1)!\) - The factorial in the denominator of the larger fraction 4. \(x^{n+1}\) - The term in the numerator of the larger fraction We will need to understand and explain the roles each of these components play in the function.
03

Rewrite the function for clarity

It is helpful to rewrite the function for better clarity. Let's rewrite the function as a single fraction: \(R_n(x) = \frac{x^{n+1}}{(1-x)^{(n+1)}(c)(n+1)!}\)
04

Explain the role of each component

1. \((1-x)^{(n+1)}\) - This term will determine the behavior of the function as x approaches 1. When x gets closer to 1, the denominator gets closer to 0, causing the function to have an infinite limit at that point. 2. \(c\) - This constant serves as a scaling factor that determines the steepness of the function with respect to x. 3. \((n+1)!\) - This term grows as n increases, serving as a balance between the rising power of x and the growth of the factorial. 4. \(x^{n+1}\) - This term serves as the basis of the series expansion, representing consecutive powers of x in the Taylor series approximation. By understanding each component, we now have a better understanding of the function.

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 Expansion
In mathematics, series expansion is a powerful tool that allows complex functions to be expressed in simpler terms. It breaks down a function into a sum of its terms. When we talk about Taylor series, we're referring to one specific type of series expansion. A Taylor series tries to approximate a function by using infinite sums of its derivatives at a single point.

For example, if you have a function you want to analyze around a particular point, you can express it as an infinite sum of terms. Each term is calculated using the derivatives of the function at that point, typically around zero, known as the Maclaurin series, or around another specified point for general Taylor series.

The closer a series expansion approximates a function, the more terms you might use. So each additional term helps to improve this approximation. In practice, we often truncate the series to a finite number of terms to suit practical calculations, keeping in mind how many terms are necessary for an acceptable level of accuracy.
Factorial Function
The factorial function, denoted as (n!) , is an important concept in various mathematical fields. It refers to the product of all positive integers up to a given number. For example, 4! equals 4 脳 3 脳 2 脳 1 = 24. Factorials play a crucial role in combinations, permutations, and other calculations involving counting.

They are especially significant in series expansions like the Taylor series. Here, factorials help regulate the magnitude of terms as the exponent of each term in the expansion grows in response to an increase in index. The term ((n+1)!) in the denominator of the function affects how rapidly or slowly the series grows relative to x depending on the value of n.

More specifically, factorials often serve to "balance" the results of the computations, ensuring that each term makes a sensible and manageable contribution to the overall series. This is why as n increases, the value of (n!) becomes significant; it plays a critical balancing role especially for larger x values.
Function Behavior
Understanding function behavior is all about investigating how a function acts or changes under different conditions. For the given function, the behavior is influenced by all of its components, particularly as x approaches specific values, like 1 in this case.

The term ((1-x)^{(n+1)}) particularly dictates the function's behavior near x = 1 . As x approaches 1, this component makes the denominator shrink towards zero, often causing the function to head towards infinity. This is an example of a vertical asymptote, which is crucial in understanding the limits and behavior of the function near critical points.

In addition, the constant c impacts the function's steepness or rate of change. Essentially, it acts like a multiplier to scale the function up or down. When learning about these elements, it is important to recognize their individual roles.
  • Examine how various values of x affect the numerator and denominator separately.
  • Consider the impact of changing values on the overall behavior.
  • Understand scaling through constants like to predict changes in graph steepness.
Observing these factors will significantly aid in fully grasping how the function behaves across its domain.

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 remainder in the Taylor series centered at the point a for the following functions. Then show that \(\lim _{n \rightarrow \infty} R_{n}(x)=0\) for all \(x\) in the interval of convergence. $$f(x)=\cos 2 x, a=0$$

a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. You do not need to use the definition of the Taylor series coefficients. b. Determine the radius of convergence of the series. $$f(x)=\frac{e^{x}+e^{-x}}{2}$$

Exponential function In Section 9.3, we show that the power series for the exponential function centered at 0 is $$e^{x}=\sum_{k=0}^{\infty} \frac{x^{k}}{k !}, \quad \text { for }-\infty< x <\infty$$ Use the methods of this section to find the power series for the following functions. Give the interval of convergence for the resulting series. $$f(x)=x^{2} e^{x}$$

Show that the first five nonzero coefficients of the Taylor series (binomial series) for \(f(x)=\sqrt{1+4 x}\) abou 0 are integers. (In fact, all the coefficients are integers.)

Match functions a-f with Taylor polynomials A-F (all centered at 0). Give reasons for your choices. a. \(\sqrt{1+2 x} \quad\) A. \(p_{2}(x)=1+2 x+2 x^{2}\) b. \(\frac{1}{\sqrt{1+2 x}}\quad\) B. \(p_{2}(x)=1-6 x+24 x^{2}\) c. \(e^{2 x}\) \(\quad\) C. \(p_{2}(x)=1+x-\frac{x^{2}}{2}\) d. \(\frac{1}{1+2 x}\) \(\quad\) D. \(p_{2}(x)=1-2 x+4 x^{2}\) e. \(\frac{1}{(1+2 x)^{3}}\) \(\quad\) E. \(p_{2}(x)=1-x+\frac{3}{2} x^{2}\) f. \(e^{-2 x}\) \(\quad\) F. \(p_{2}(x)=1-2 x+2 x^{2}\)
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Calculus

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.