Textbook Question
Taylor series and interval of convergence
b. Write the power series using summation notation.
f(x) = cosh 3x, a = 0
Ch. 11 - Power Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Not the one you use?Change textbookSelect a different textbook
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 11.4.78b
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.
b. Integrate the series to find a Taylor series for Si.
Verified step by step guidance1
Recall that the sine integral function is defined as \(\mathrm{Si}(x) = \int_0^x f(t) \, dt\), where \(f(t) = \frac{\sin t}{t}\) for \(t \neq 0\) and \(f(0) = 1\).
Start by expressing \(\sin t\) as its Taylor series expansion around \(t=0\):
\(\sin t = \sum_{n=0}^\infty (-1)^n \frac{t^{2n+1}}{(2n+1)!}\).
Divide the series for \(\sin t\) by \(t\) to get the series for \(f(t) = \frac{\sin t}{t}\):
\(f(t) = \sum_{n=0}^\infty (-1)^n \frac{t^{2n}}{(2n+1)!}\).
Integrate the series term-by-term from 0 to \(x\) to find the Taylor series for \(\mathrm{Si}(x)\):
\(\mathrm{Si}(x) = \int_0^x f(t) \, dt = \int_0^x \sum_{n=0}^\infty (-1)^n \frac{t^{2n}}{(2n+1)!} \, dt\).
Interchange the integral and the summation (justified by uniform convergence on compact intervals) and integrate each term:
\(\mathrm{Si}(x) = \sum_{n=0}^\infty (-1)^n \frac{1}{(2n+1)!} \int_0^x t^{2n} \, dt = \sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{(2n+1)(2n+1)!}\).
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
6mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definition and Properties of the Sine Integral Function
The sine integral function Si(x) is defined as the integral from 0 to x of (sin t)/t dt, with a special value at t=0 to ensure continuity. Understanding this function involves recognizing it as an integral of a function with a removable discontinuity and its role in analysis and applications.
Recommended video:
05:43
Definition of the Definite Integral
Taylor Series Expansion of Functions
A Taylor series represents a function as an infinite sum of terms calculated from the derivatives at a single point, usually zero. To find the Taylor series of Si(x), one must express the integrand as a power series and then integrate term-by-term, ensuring convergence within the radius of expansion.
Recommended video:
08:42Taylor Series
Term-by-Term Integration of Power Series
Integrating a power series term-by-term is valid within the interval of convergence and allows finding the integral of complex functions by integrating each term individually. This technique simplifies finding the Taylor series of integral-defined functions like Si(x) by integrating the series expansion of (sin t)/t.
Recommended video:
05:58Intro to Power Series
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.
b. Write the power series using summation notation.
f(x) = 1/x², a=1
Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
b. The function f(x) = csc x has a Taylor series centered at π/2.
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) =√(1+x) ≈ 1 + x/2
Textbook Question
Taylor series
b. Write the power series using summation notation.
f(x)=sin x, a = π/2
1
views
Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
b. Let f(x)=x⁵−1 The Taylor polynomial for f of order 10 centered at 0 is f itself.