Textbook Question
Intervals of Convergence
In Exercises 1–36, (a) find the series’ radius and interval of convergence. For what values of x does the series converge (b) absolutely, (c) conditionally?
∑ (from n = 1 to ∞) [ (ln n) xⁿ ]
Ch. 10 - Infinite Sequences and Series
Hass15th EditionThomas' CalculusISBN: 9780137616077
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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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- 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 \(x\).
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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.
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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.
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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.
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Related Practice
Textbook Question
In Exercises 43–50, use Theorem 20 to find the series’ interval of convergence and, within this interval, the sum of the series as a function of x.
∑ (from n = 0 to ∞) [ (x² − 1) / 2 ]ⁿ
Textbook Question
Intervals of Convergence
In Exercises 1–36, (a) find the series’ radius and interval of convergence. For what values of x does the series converge (b) absolutely, (c) conditionally?
∑ (from n = 0 to ∞) (x + 5)ⁿ
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
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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
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Textbook Question
Use series to evaluate the limits in Exercises 29–40.
29. lim (x → 0) (e^x - (1 + x)) / x²