Textbook Question
Limits with a parameter Use Taylor series to evaluate the following limits. Express the result in terms of the nonzero real parameter(s).
lim ₓ→₀ (eᵃˣ − 1)/x
Ch. 11 - Power Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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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 11.2.57
Functions to power series Find power series representations centered at 0 for the following functions using known power series. Give the interval of convergence for the resulting series.
f(x) = 2x/(1 + x²)²
Verified step by step guidance1
Recall the known power series for the function \( \frac{1}{1 - t} = \sum_{n=0}^{\infty} t^n \) for \( |t| < 1 \). This is a geometric series centered at 0.
Recognize that \( \frac{1}{(1 + x^2)^2} \) can be related to the derivative of \( \frac{1}{1 + x^2} \). Start with \( \frac{1}{1 + x^2} = \sum_{n=0}^{\infty} (-1)^n x^{2n} \) for \( |x| < 1 \).
Differentiate the series term-by-term to find the series for \( \frac{d}{dx} \left( \frac{1}{1 + x^2} \right) = \frac{d}{dx} \left( \sum_{n=0}^{\infty} (-1)^n x^{2n} \right) \). This gives \( \frac{-2x}{(1 + x^2)^2} = \sum_{n=0}^{\infty} (-1)^n 2n x^{2n-1} \).
Multiply both sides of the differentiated series by \(-1\) to isolate \( \frac{2x}{(1 + x^2)^2} \), which matches the function \( f(x) \). So, \( f(x) = 2x/(1 + x^2)^2 = \sum_{n=0}^{\infty} (-1)^{n+1} 2n x^{2n-1} \).
State the interval of convergence for the power series, which remains \( |x| < 1 \) because the operations of differentiation and multiplication by \( x \) do not change the radius of convergence.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Power Series Representation
A power series is an infinite sum of terms in the form a_n(x - c)^n, where c is the center of the series. Representing functions as power series allows approximation and analysis using polynomials. Finding a power series centered at 0 means expressing the function as a sum involving powers of x.
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Intro to Power Series
Known Power Series and Manipulation
Using standard power series expansions, such as for 1/(1 - x) or 1/(1 + x^2), helps derive new series by algebraic operations like differentiation, multiplication, or substitution. Recognizing and manipulating these known series is key to finding the series for more complex functions.
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05:58Intro to Power Series
Interval of Convergence
The interval of convergence is the set of x-values for which the power series converges to the function. Determining this interval involves applying convergence tests, such as the ratio or root test, and is essential to understand where the series accurately represents the function.
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08:44Interval of Convergence
Related Practice
Textbook Question
Limits Evaluate the following limits using Taylor series.
lim ₓ→₀ (sin 2x)/x
Textbook Question
Derivative trick Here is an alternative way to evaluate higher derivatives of a function f that may save time. Suppose you can find the Taylor series for f centered at the point a without evaluating derivatives (for example, from a known series). Then f⁽ᵏ⁾(a)=k! multiplied by the coefficient of (x−a)ᵏ. Use this idea to evaluate f⁽³⁾(0) and f⁽⁴⁾(0) for the following functions. Use known series and do not evaluate derivatives.
f(x) = eᶜᵒˢ ˣ
Textbook Question
{Use of Tech} Approximating definite integrals Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than 10⁻⁴.
∫₀⁰ᐧ³⁵ tan ⁻¹x dx
Textbook Question
A limit by Taylor series Use Taylor series to evaluate lim ₓ→₀ ((sin x)/x)¹/ˣ²
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Textbook Question
Radius and interval of convergence Determine the radius and interval of convergence of the following power series.
∑ₖ₌₀∞ (-x/10)²ᵏ