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 11.4.79b
{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.
Verified step by step guidance1
Recall the Maclaurin series expansions for sine and cosine functions: \( \sin z = \sum_{n=0}^\infty (-1)^n \frac{z^{2n+1}}{(2n+1)!} \) and \( \cos z = \sum_{n=0}^\infty (-1)^n \frac{z^{2n}}{(2n)!} \).
Substitute \( z = t^2 \) into these series to get the expansions for \( \sin t^2 \) and \( \cos t^2 \):
\[ \sin t^2 = \sum_{n=0}^\infty (-1)^n \frac{t^{4n+2}}{(2n+1)!} \quad \text{and} \quad \cos t^2 = \sum_{n=0}^\infty (-1)^n \frac{t^{4n}}{(2n)!} \].
Write out the first four nonzero terms explicitly for both series:
For \( \sin t^2 \):
\[ t^2 - \frac{t^6}{3!} + \frac{t^{10}}{5!} - \frac{t^{14}}{7!} + \cdots \]
For \( \cos t^2 \):
\[ 1 - \frac{t^4}{2!} + \frac{t^8}{4!} - \frac{t^{12}}{6!} + \cdots \].
Integrate each term of the series for \( \sin t^2 \) from 0 to \( x \) to find the Maclaurin series for \( S(x) = \int_0^x \sin t^2 \, dt \):
\[ \int_0^x t^{m} dt = \frac{x^{m+1}}{m+1} \], so integrate term-by-term accordingly.
Similarly, integrate each term of the series for \( \cos t^2 \) from 0 to \( x \) to find the Maclaurin series for \( C(x) = \int_0^x \cos t^2 \, dt \), again using term-by-term integration and the power rule for integrals.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Fresnel Integrals
Fresnel integrals S(x) and C(x) are defined as integrals of sine and cosine functions with quadratic arguments, specifically S(x) = ∫₀ˣ sin(t²) dt and C(x) = ∫₀ˣ cos(t²) dt. They arise in optics and wave theory, describing diffraction patterns and wave propagation.
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Integration by Parts for Definite Integrals
Maclaurin Series Expansion
A Maclaurin series is a Taylor series expansion of a function about zero. It expresses functions as infinite sums of powers of the variable, with coefficients derived from derivatives at zero. Expanding sin(t²) and cos(t²) into Maclaurin series involves substituting t² into the known series for sine and cosine.
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Term-by-Term Integration of Power Series
Once sin(t²) and cos(t²) are expanded into power series, integrating term-by-term from 0 to x is valid under uniform convergence. This process yields the Maclaurin series for S(x) and C(x) by integrating each power of t individually, resulting in a series representation for the Fresnel integrals.
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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
Taylor series and interval of convergence
b. Write the power series using summation notation.
f(x) = (1 + x²)⁻¹, a = 0
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.
b. Estimate f(0.2) and give a bound on the error in the approximation.
f(x) = sin x ≈ x
Textbook Question
Taylor series
b. Write the power series using summation notation.
f(x) = ln x, a = 3