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 11.3.25a
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
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, \(f(x) = \ln(x - 2)\) and the center is \(a = 3\). We will find derivatives of \(f\) at \(x=3\).
Compute the first derivative:
\[f'(x) = \frac{1}{x - 2}.\]
Evaluate at \(x=3\):
\[f'(3) = \frac{1}{3 - 2} = 1.\]
Find higher order derivatives by differentiating repeatedly:
- Second derivative:
\[f''(x) = -\frac{1}{(x - 2)^2}\]
Evaluate at \(x=3\):
\[f''(3) = -1.\]
- Third derivative:
\[f^{(3)}(x) = \frac{2}{(x - 2)^3}\]
Evaluate at \(x=3\):
\[f^{(3)}(3) = 2.\]
- Fourth derivative:
\[f^{(4)}(x) = -\frac{6}{(x - 2)^4}\]
Evaluate at \(x=3\):
\[f^{(4)}(3) = -6.\]
Write the first four nonzero terms of the Taylor series using the formula:
\[f(x) \approx f(3) + f'(3)(x - 3) + \frac{f''(3)}{2!}(x - 3)^2 + \frac{f^{(3)}(3)}{3!}(x - 3)^3 + \frac{f^{(4)}(3)}{4!}(x - 3)^4.\]
Substitute the values found for \(f(3)\) and the derivatives to express the series up to the fourth term.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
7mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Taylor and Maclaurin Series
A Taylor series represents a function as an infinite sum of terms calculated from the derivatives of the function at a single point a. When a = 0, it is called a Maclaurin series. Each term involves the nth derivative evaluated at a, multiplied by (x - a)^n and divided by n!. This series approximates the function near the point a.
Recommended video:
08:26
Convergence of Taylor & Maclaurin Series
Derivatives of Logarithmic Functions
To find the Taylor series of f(x) = ln(x - 2), you need to compute successive derivatives of the logarithmic function. The first derivative is 1/(x - 2), and higher derivatives involve powers of (x - 2) in the denominator with alternating signs. Understanding these derivatives is essential to form the terms of the series.
Recommended video:
05:18Derivative of the Natural Logarithmic Function
Interval of Convergence
The interval of convergence is the range of x-values for which the Taylor series converges to the function. For ln(x - 2) centered at a = 3, the series converges where |x - 3| is less than the distance to the nearest singularity (x = 2). Determining this interval ensures the series accurately represents the function.
Recommended video:
08:44Interval of Convergence
Related Practice
Textbook Question
{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.
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) = e²ˣ, a = 0
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
{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).