Textbook Question
Taylor series and interval of convergence
b. Write the power series using summation notation.
f(x) = e²ˣ, a = 0
41
views
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 11.3.31a
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) = ln x, 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, \(f(x) = \ln x\) and \(a = 3\). We will need to find the derivatives of \(f(x)\) evaluated at \(x=3\).
Calculate the first four derivatives of \(f(x) = \ln x\):
- \(f(x) = \ln x\)
- \(f'(x) = \frac{1}{x}\)
- \(f''(x) = -\frac{1}{x^2}\)
- \(f^{(3)}(x) = \frac{2}{x^3}\)
- \(f^{(4)}(x) = -\frac{6}{x^4}\)
Evaluate each derivative at \(x = 3\):
- \(f(3) = \ln 3\)
- \(f'(3) = \frac{1}{3}\)
- \(f''(3) = -\frac{1}{9}\)
- \(f^{(3)}(3) = \frac{2}{27}\)
- \(f^{(4)}(3) = -\frac{6}{81} = -\frac{2}{27}\)
Write the first four nonzero terms of the Taylor series using the formula:
\[f(x) \approx f(3) + f'(3)(x-3) + \frac{f''(3)}{2!}(x-3)^2 + \frac{f^{(3)}(3)}{3!}(x-3)^3 + \frac{f^{(4)}(3)}{4!}(x-3)^4,\]
substituting the values found for each derivative.
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 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 a. 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:
08:42
Taylor Series
Derivatives of the Natural Logarithm Function
To find the Taylor series of f(x) = ln(x), you need to compute its derivatives at x = a. The first derivative is 1/x, and higher derivatives follow a pattern involving factorials and powers of x. Understanding these derivatives is essential to form the series terms.
Recommended video:
05:18Derivative of the Natural Logarithmic Function
Constructing the First Four Nonzero Terms
After finding the derivatives at a, substitute them into the Taylor series formula to write out the first four nonzero terms. This involves careful calculation of each term’s coefficient and power of (x - a), ensuring the series accurately approximates ln(x) near x = 3.
Recommended video:
11:22The First Derivative Test: Finding Local Extrema Example 3
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.
f(x) = ln (x − 2), a = 3
75
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
37
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) = (1 + x²)⁻¹, a = 0
87
views
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).
46
views
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.
b. Integrate the series to find a Taylor series for Si.
120
views