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Ch. 11 - Power Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 11 - Power SeriesProblem 11.3.41
Chapter 11, Problem 11.3.41

Manipulating Taylor series Use the Taylor series in Table 11.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0.


(1 + x⁴)⁻¹

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Recall the Taylor series for the function \( \frac{1}{1 - x} = \sum_{n=0}^{\infty} x^n = 1 + x + x^2 + x^3 + \cdots \), which is valid for \( |x| < 1 \).
Notice that the given function \( (1 + x^4)^{-1} \) can be rewritten as \( \frac{1}{1 - (-x^4)} \). This means we can use the geometric series formula by substituting \( -x^4 \) in place of \( x \).
Substitute \( -x^4 \) into the geometric series to get the Taylor series: \[ \frac{1}{1 + x^4} = \sum_{n=0}^{\infty} (-x^4)^n = \sum_{n=0}^{\infty} (-1)^n x^{4n} = 1 - x^4 + x^8 - x^{12} + \cdots \]
Identify the first four nonzero terms from the series expansion: these are the terms corresponding to \( n = 0, 1, 2, 3 \), which are \( 1, -x^4, x^8, -x^{12} \).
Write the final Taylor series approximation centered at 0 using these four terms: \[ 1 - x^4 + x^8 - x^{12} \]. This is the first four nonzero terms of the Taylor series for \( (1 + x^4)^{-1} \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Taylor Series Expansion

A Taylor series represents a function as an infinite sum of terms calculated from the derivatives at a single point, usually centered at zero (Maclaurin series). It approximates functions using polynomials, making complex functions easier to analyze and manipulate.
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Taylor Series

Manipulating Known Series

Using known Taylor series expansions, such as for (1 - x)⁻¹, allows us to substitute expressions and find series for more complex functions. By replacing variables and adjusting terms, we can derive new series without computing derivatives from scratch.
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Geometric Series

Identifying Nonzero Terms

When expanding a function into a Taylor series, it is important to recognize which terms are nonzero to write the first few significant terms. This involves understanding the powers of x that appear and simplifying the series accordingly.
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Divergence Test (nth Term Test)
Related Practice
Textbook Question

Limits Evaluate the following limits using Taylor series.

lim ₓ→∞ x(e¹/ˣ − 1)

Textbook Question

Functions to power series Find power series representations centered at 0 for the following functions using known power series. Give the interval of convergence for the resulting series.

f(x) = ln √(4 − x²)

Textbook Question

{Use of Tech} Best center point Suppose you wish to approximate cos (π/ 2) using Taylor polynomials. Is the approximation more accurate if you use Taylor polynomials centered at 0 or at π/6? Use a calculator for numerical experiments and check for consistency with Theorem 11.2. Does the answer depend on the order of the polynomial?

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Textbook Question

Power series for derivatives


a. Differentiate the Taylor series centered at 0 for the following functions.

b. Identify the function represented by the differentiated series.

c. Give the interval of convergence of the power series for the derivative.


f(x) = (1 − x)⁻¹

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Textbook Question

Radius and interval of convergence Determine the radius and interval of convergence of the following power series.


∑ₖ₌₁∞ (kx)ᵏ

Textbook Question

Suppose f(0)=1, f'(0)=0, f''(0)=2, and f⁽³⁾(0)=6. Find the third-order Taylor polynomial for f centered at 0 and use it to approximate f(0.2).