Textbook Question
Taylor series
a. Use the definition of a Taylor series to find the first four nonzero terms of the Taylor series for the given function centered at a.
f(x) = 1/x, a = 1
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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 11.3.49a
{Use of Tech} Binomial series
a. Find the first four nonzero terms of the binomial series centered at 0 for the given function.
f(x) = (1+x)⁻²/³; approximate 1.18⁻²/³.
Verified step by step guidance1
Identify the function and the form of the binomial series: The function is \(f(x) = (1+x)^{-\frac{2}{3}}\). The binomial series expansion for \((1+x)^k\) centered at 0 is given by \(\sum_{n=0}^{\infty} \binom{k}{n} x^n\), where \(\binom{k}{n} = \frac{k(k-1)(k-2)\cdots(k-n+1)}{n!}\).
Calculate the binomial coefficients for the first four terms: Start with \(n=0\), where \(\binom{k}{0} = 1\). Then find \(\binom{-\frac{2}{3}}{1}\), \(\binom{-\frac{2}{3}}{2}\), and \(\binom{-\frac{2}{3}}{3}\) using the formula for generalized binomial coefficients.
Write out the first four nonzero terms of the series: Using the coefficients found, express the terms as \(\binom{k}{n} x^n\) for \(n=0,1,2,3\). This will give you the approximation of \(f(x)\) near \(x=0\).
Use the series to approximate \(f(0.18)\): Since \(1.18 = 1 + 0.18\), substitute \(x=0.18\) into the first four terms of the binomial series to approximate \(1.18^{-\frac{2}{3}}\).
Sum the four terms to get the approximate value: Add the four terms calculated in the previous step to find the approximate value of \(1.18^{-\frac{2}{3}}\) using the binomial series expansion.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Binomial Series Expansion
The binomial series generalizes the binomial theorem to any real exponent, allowing the expansion of expressions like (1 + x)^k into an infinite series. It is expressed as a sum involving binomial coefficients and powers of x, converging for |x| < 1. This expansion helps approximate functions near x = 0.
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Geometric Series
Finding Nonzero Terms in a Series
When expanding a function into a series, identifying the first few nonzero terms involves calculating coefficients using the binomial formula and recognizing when terms vanish. This process is essential for approximations and understanding the behavior of the function near the center of expansion.
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Using Series to Approximate Function Values
Once the binomial series is found, it can be used to approximate function values at points close to the center by substituting the value into the partial sum of the series. This method provides an efficient way to estimate values like 1.18^(-2/3) without a calculator.
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Related Practice
Textbook Question
Taylor series and interval of convergence
a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a.
f(x) = ln (x − 2), a = 3
Textbook Question
Taylor series
a. Use the definition of a Taylor series to find the first four nonzero terms of the Taylor series for the given function centered at a.
f(x)=sin x, a = π/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) = eˣ ≈ 1 + x
Textbook Question
Taylor series and interval of convergence
a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a.
f(x) = cosh 3x, a = 0
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
a. Compute S′(x) and C′(x).