Textbook Question
Taylor series and interval of convergence
b. Write the power series using summation notation.
f(x)=2/(1−x)³, a=0
Ch. 11 - Power Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 11, Problem 11.3.15b
Taylor series and interval of convergence
b. Write the power series using summation notation.
f(x) = (1 + x²)⁻¹, a = 0
Verified step by step guidance1
Recognize that the function given is \( f(x) = (1 + x^2)^{-1} \), which resembles the form of a geometric series \( \frac{1}{1 - r} = \sum_{n=0}^{\infty} r^n \) when \( |r| < 1 \).
Rewrite the function to match the geometric series form by identifying \( r = -x^2 \), so \( f(x) = \frac{1}{1 - (-x^2)} \).
Express the power series as a summation using the geometric series formula: \[ f(x) = \sum_{n=0}^{\infty} (-x^2)^n \].
Simplify the term inside the summation to get \( (-1)^n x^{2n} \), so the power series becomes \[ f(x) = \sum_{n=0}^{\infty} (-1)^n x^{2n} \].
Note that the interval of convergence is determined by \( |r| < 1 \), which means \( |-x^2| = |x|^2 < 1 \), so the interval of convergence is \( |x| < 1 \).
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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 of the function at a single point. For a function f(x) centered at a = 0, it is expressed as f(x) = Σ (f⁽ⁿ⁾(0)/n!) xⁿ, where f⁽ⁿ⁾(0) is the nth derivative evaluated at 0. This allows approximation of functions using polynomials.
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Taylor Series
Power Series and Summation Notation
A power series is an infinite series of the form Σ cₙ (x - a)ⁿ, where cₙ are coefficients and a is the center. Summation notation compactly expresses this infinite sum, making it easier to manipulate and analyze. Writing a function as a power series involves finding the coefficients cₙ that match the function's behavior.
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Interval of Convergence
The interval of convergence is the set of x-values for which a power series converges to the function. It is determined by testing the radius of convergence, often using the ratio or root test. Understanding this interval is crucial to knowing where the power series accurately represents the function.
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Related Practice
Textbook Question
Symmetry
b. Use infinite series to show that sin x is an odd function. That is, show sin (-x) = -sin x.
Textbook Question
{Use of Tech} Fresnel integrals The theory of optics gives rise to the two Fresnel integrals
S(x) = ∫₀ˣ sin t² dt and C(x) = ∫₀ˣ cos t² dt
b. Expand sin t² and cos t² in a Maclaurin series, and then integrate to find the first four nonzero terms of the Maclaurin series for S and C.
Textbook Question
{Use of Tech} Small argument approximations Consider the following common approximations when x is near zero.
b. Estimate f(0.2) and give a bound on the error in the approximation.
f(x) = ln (1 + x) ≈ x − x²/2
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Textbook Question
{Use of Tech} Small argument approximations Consider the following common approximations when x is near zero.
a. Estimate f(0.1) and give a bound on the error in the approximation.
f(x) = tan⁻¹ x ≈ x
Textbook Question
Taylor series
b. Write the power series using summation notation.
f(x) = ln x, a = 3