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.
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 11.3.17a
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) = e²ˣ, a = 0
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 the center: here, \(f(x) = e^{2x}\) and \(a = 0\), so we are looking for the Maclaurin series (Taylor series at 0).
Compute the derivatives of \(f(x)\) and evaluate them at \(x=0\):
- \(f(x) = e^{2x}\)
- \(f'(x) = 2e^{2x}\)
- \(f''(x) = 4e^{2x}\)
- \(f^{(3)}(x) = 8e^{2x}\)
Evaluate each at \(x=0\):
- \(f(0) = e^0 = 1\)
- \(f'(0) = 2e^0 = 2\)
- \(f''(0) = 4e^0 = 4\)
- \(f^{(3)}(0) = 8e^0 = 8\)
Write the first four nonzero terms of the Taylor series using the formula:
\[f(x) \approx f(0) + \frac{f'(0)}{1!} x + \frac{f''(0)}{2!} x^2 + \frac{f^{(3)}(0)}{3!} x^3,\]
and substitute the values found in the previous step.
Simplify each term by dividing the derivative values by the factorials and write the series explicitly up to the \(x^3\) term.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Taylor and Maclaurin 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. When centered at zero, it is called a Maclaurin series. Each term involves the nth derivative evaluated at the center, multiplied by (x - a)^n and divided by n!.
Recommended video:
08:26
Convergence of Taylor & Maclaurin Series
Derivatives of Exponential Functions
The exponential function e^(kx) has derivatives that are proportional to itself, specifically the nth derivative is k^n * e^(kx). This property simplifies finding terms in the Taylor series since each derivative at the center can be easily computed.
Recommended video:
04:50Derivatives of General Exponential Functions
Interval of Convergence
The interval of convergence is the set of x-values for which the Taylor series converges to the function. For exponential functions like e^(2x), the series converges for all real x, meaning the interval of convergence is (-∞, ∞).
Recommended video:
08:44Interval of Convergence
Related Practice
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
1
views
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 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) = (1 + x²)⁻¹, a = 0
Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
The interval of convergence of the power series ∑ cₖ(x−3)ᵏ could be (−2,8).
1
views
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).