Textbook Question
Finding Taylor and Maclaurin Series
In Exercises 25–34, find the Taylor series generated by f at x = a.
f(x) = x³ − 2x + 4,a = 2
1
views
Ch. 10 - Infinite Sequences and Series
Hass15th EditionThomas' CalculusISBN: 9780137616077
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 10, Problem 10.9.2
Finding Taylor Series
Use substitution (as in Formula (7)) to find the Taylor series at x = 0 of the functions in Exercises 1–12.
e⁻ˣ/²
Verified step by step guidance1
Recall the Taylor series expansion of the exponential function \(e^x\) centered at \(x=0\), which is given by:
\[e^x = \sum_{n=0}^\infty \frac{x^n}{n!}\]
In the given function, we have \(e^{-x^2}\). To use substitution, let’s replace \(x\) in the exponential series with \(-x^2\). This gives:
\[e^{-x^2} = \sum_{n=0}^\infty \frac{(-x^2)^n}{n!}\]
Simplify the powers inside the summation:
\[e^{-x^2} = \sum_{n=0}^\infty \frac{(-1)^n x^{2n}}{n!}\]
This series is already centered at \(x=0\), so this is the Taylor series expansion of \(e^{-x^2}\) at \(x=0\).
To write out the first few terms explicitly, substitute \(n=0,1,2,3,\ldots\) into the summation to see the pattern of coefficients and powers of \(x\).
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 Expansion
A Taylor series represents a function as an infinite sum of terms calculated from the derivatives at a single point, usually around x = 0 (Maclaurin series). It approximates functions using polynomials, making complex functions easier to analyze and compute.
Recommended video:
08:42
Taylor Series
Substitution Method in Series
Substitution involves replacing the variable in a known Taylor series with another expression to find the series of a related function. For example, substituting x² into the series for e^x helps find the series for e^(x²). This method simplifies finding series for composite functions.
Recommended video:
06:00Geometric Series
Exponential Function Series
The exponential function e^x has a well-known Taylor series: sum of x^n/n! for n from 0 to infinity. Understanding this base series is essential because many functions, like e^(-x²/2), can be expressed by modifying this series through substitution and algebraic manipulation.
Recommended video:
6:13Exponential Functions
Related Practice
Textbook Question
Is it true that a sequence {aₙ} of positive numbers must converge if it is bounded above? Give reasons for your answer.
1
views
Textbook Question
Finding Taylor Polynomials
In Exercises 1–10, find the Taylor polynomials of orders 0, 1, 2, and 3 generated by f at a.
f(x) = sin x,a = 0
1
views
Textbook Question
Use series to evaluate the limits in Exercises 29–40.
29. lim (x → 0) (e^x - (1 + x)) / x²
Textbook Question
Determining Convergence or Divergence
Which of the series in Exercises 13–46 converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)
∑ (from n=1 to ∞) eⁿ / (1 + e²ⁿ)
1
views
Textbook Question
In Exercises 57–82, use any method to determine whether the series converges or diverges. Give reasons for your answer.
∑ (from n = 2 to ∞) [(ln n / n)³]