Textbook Question
Sine integral function The function Si(x) = ∫₀ˣ f(t) dt, where f(t) = {(sin t)/t if t ≠ 0, 1 if t = 0, is called the sine integral function.
a. Expand the integrand in a Taylor series centered at 0.
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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 11.3.29a
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
Verified step by step guidance1
Recall the definition of the Taylor series of a function \(f(x)\) centered at \(a\):
\[f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n,\]
where \(f^{(n)}(a)\) is the \(n\)-th derivative of \(f\) evaluated at \(x = a\).
Identify the function and center: here, \(f(x) = \frac{1}{x}\) and \(a = 1\). We will need to find the derivatives of \(f(x)\) evaluated at \(x=1\).
Compute the first four derivatives of \(f(x)\):
- \(f(x) = x^{-1}\)
- \(f'(x) = -x^{-2}\)
- \(f''(x) = 2x^{-3}\)
- \(f^{(3)}(x) = -6x^{-4}\)
- \(f^{(4)}(x) = 24x^{-5}\)
Evaluate each derivative at \(x = 1\):
- \(f(1) = 1\)
- \(f'(1) = -1\)
- \(f''(1) = 2\)
- \(f^{(3)}(1) = -6\)
- \(f^{(4)}(1) = 24\)
Write the first four nonzero terms of the Taylor series using the formula:
\[f(x) \approx f(1) + \frac{f'(1)}{1!}(x-1) + \frac{f''(1)}{2!}(x-1)^2 + \frac{f^{(3)}(1)}{3!}(x-1)^3,\]
substituting the values found for each derivative.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Taylor Series Definition
A Taylor series represents a function as an infinite sum of terms calculated from the derivatives of the function at a single point, called the center. Each term involves the nth derivative evaluated at the center, multiplied by (x - a)^n and divided by n!. This series approximates the function near the center point.
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Derivatives of the Function
To construct the Taylor series, you need to compute successive derivatives of the function at the center point. For f(x) = 1/x, derivatives involve powers of x with alternating signs. Evaluating these derivatives at a = 1 provides the coefficients for the series terms.
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Constructing the Series Terms
Each term of the Taylor series is formed by dividing the nth derivative at the center by n! and multiplying by (x - a)^n. Identifying the first four nonzero terms requires calculating derivatives up to the third order and substituting into this formula to write the polynomial approximation.
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Related Practice
Textbook Question
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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) = e²ˣ, a = 0
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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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{Use of Tech} Bessel functions Bessel functions arise in the study of wave propagation in circular geometries (for example, waves on a circular drum head). They are conveniently defined as power series. One of an infinite family of Bessel functions is
J₀(x) = ∑ₖ₌₀∞ (−1)ᵏ/(2²ᵏ(k!)²) x²ᵏ
a. Write out the first four terms of J₀.
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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.
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{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).
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