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Chapter 9: Problem 17

Find the Maclaurin polynomial of order \(n\) for \(f(x)=1 /(1-x)\). Then use it with \(n=4\) to approximate each of the following. (a) \(f(0.1)\) (b) \(f(0.5)\) (c) \(f(0.9)\) (d) \(f(2)\)

Short Answer

Expert verified
(a) 1.1111, (b) 1.9375, (c) 4.0951, (d) Not applicable (series diverges for x 鈮� 1)

Step by step solution

01

Understand the function and Maclaurin series

The Maclaurin series is a Taylor series expansion of a function about 0. For the function \( f(x) = \frac{1}{1-x} \), the Maclaurin series is given by \( 1 + x + x^2 + x^3 + \cdots \) which can be generalized to a polynomial of order \( n \) as \( T_n(x) = 1 + x + x^2 + \cdots + x^n \).
02

Find the 4th order Maclaurin polynomial

Using the general form from Step 1, the 4th order Maclaurin polynomial for \( f(x) = \frac{1}{1-x} \) is:\[ T_4(x) = 1 + x + x^2 + x^3 + x^4 \]
03

Approximate f(0.1) using T_4(x)

Substitute \( x = 0.1 \) into the polynomial:\[ T_4(0.1) = 1 + 0.1 + 0.1^2 + 0.1^3 + 0.1^4 \]Calculate each term:- \( 0.1^2 = 0.01 \)- \( 0.1^3 = 0.001 \)- \( 0.1^4 = 0.0001 \)Thus:\[ T_4(0.1) = 1 + 0.1 + 0.01 + 0.001 + 0.0001 = 1.1111 \]
04

Approximate f(0.5) using T_4(x)

Substitute \( x = 0.5 \) into the polynomial:\[ T_4(0.5) = 1 + 0.5 + 0.5^2 + 0.5^3 + 0.5^4 \]Calculate each term:- \( 0.5^2 = 0.25 \)- \( 0.5^3 = 0.125 \)- \( 0.5^4 = 0.0625 \)Thus:\[ T_4(0.5) = 1 + 0.5 + 0.25 + 0.125 + 0.0625 = 1.9375 \]
05

Approximate f(0.9) using T_4(x)

Substitute \( x = 0.9 \) into the polynomial:\[ T_4(0.9) = 1 + 0.9 + 0.9^2 + 0.9^3 + 0.9^4 \]Calculate each term:- \( 0.9^2 = 0.81 \)- \( 0.9^3 = 0.729 \)- \( 0.9^4 = 0.6561 \)Thus:\[ T_4(0.9) = 1 + 0.9 + 0.81 + 0.729 + 0.6561 = 4.0951 \]
06

Understand shortcoming for approximating f(2)

The series \( 1 + x + x^2 + x^3 + \cdots \) converges for \( |x| < 1 \). Therefore, using this series to approximate \( f(2) \) results in divergence, as \( 2 > 1 \). The approximation is incorrect, and the polynomial doesn't approximate \( f(2) \) effectively.

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Key Concepts

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

Polynomials
A polynomial is a mathematical expression made up of variables, coefficients, and exponents, combined using addition, subtraction, and multiplication. Polynomials are like algebraic recipes that can model a variety of behaviors. For instance:
  • The function \(f(x) = 1 + x + x^2 + x^3\) is a polynomial.
  • The expressions extend to any order (or degree), denoted by the highest power of \(x\).
The Maclaurin polynomial, which is a specific type of polynomial, approximates functions near zero. When developing the Maclaurin polynomial for \(f(x) = \frac{1}{1-x}\), as seen above, it becomes \(1 + x + x^2 + \ldots + x^n\). This showcases simplicity, a staple of polynomials.
Taylor Series
The Taylor series is a powerful tool in mathematics that represents functions as infinite sums of terms. Each term in the series adds more detail to the approximation of the function. The Maclaurin series is a special case of the Taylor series centered at zero. To illustrate:
  • For \(f(x) = \frac{1}{1-x}\), the Maclaurin series expansion starts as \(1 + x + x^2 + \ldots\).
  • This series can be manipulated to different orders depending on the desired accuracy, which, for our exercise, is \(n=4\) resulting in \(T_4(x) = 1 + x + x^2 + x^3 + x^4\).
Each additional term in the series function provides a refined approximation around zero, making it versatile for functions that are otherwise difficult to compute directly.
Convergence
Convergence is a crucial concept when working with series, such as the Maclaurin or Taylor series. It refers to whether the sum of the series approaches a definite value as more terms are added. Consider these key points:
  • For a series like \(1 + x + x^2 + \ldots\), convergence occurs when \(|x| < 1\).
  • Within this range, adding terms helps the polynomial approach the function鈥檚 actual value.
  • Outside of this range, such as \(f(2)\), the series diverges and fails to produce a valid approximation.
In context, while \(T_4(x)\) performs well for values like \(f(0.1)\), \(f(0.5)\), and \(f(0.9)\), it struggles beyond \(x = 1\), highlighting both the power and limitation of convergence.

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Most popular questions from this chapter

Suppose that Mary rolls a fair die until a "6" occurs. Let \(X\) denote the random variable that is the number of tosses needed for this " 6 " to occur. Find the probability distribution for \(X\) and verify that all the probabilities sum to 1 .

Determine convergence or divergence for each of the series. Indicate the test you use. $$ \sum_{n=1}^{\infty} \frac{n}{2+n 5^{n}} $$

Find the terms through \(x^{5}\) in the Maclaurin series for \(f(x) .\) Hint: It may be easiest to use known Maclaurin series and then perform multiplications, divisions, and so on. For example, \(\tan x=(\sin x) /(\cos x) .\) \(f(x)=x \sec \left(x^{2}\right)+\sin x\)

Find the Taylor polynomial of order 4 based at 2 for \(f(x)=x^{4}\) and show that it represents \(f(x)\) exactly.

Find the terms through \(x^{5}\) in the Maclaurin series for \(f(x) .\) Hint: It may be easiest to use known Maclaurin series and then perform multiplications, divisions, and so on. For example, \(\tan x=(\sin x) /(\cos x) .\) \(f(x)=(1+x)^{3 / 2}\)
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