Textbook Question
{Use of Tech} Small argument approximations Consider the following common approximations when x is near zero.
b. Estimate f(0.2) and give a bound on the error in the approximation.
f(x) = sin x ≈ x
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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 11.3.17b
Taylor series and interval of convergence
b. Write the power series using summation notation.
f(x) = e²ˣ, a = 0
Verified step by step guidance1
Recall the Taylor series expansion of a function \( f(x) \) about \( a = 0 \) (Maclaurin series) is given by:
\[
\displaystyle f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n
\]
Identify the function \( f(x) = e^{2x} \). The derivatives of \( f(x) \) are all of the form \( f^{(n)}(x) = 2^n e^{2x} \). Evaluating at \( x = 0 \), we get:
\[
f^{(n)}(0) = 2^n e^0 = 2^n
\]
Substitute \( f^{(n)}(0) = 2^n \) into the Taylor series formula:
\[
\displaystyle e^{2x} = \sum_{n=0}^{\infty} \frac{2^n}{n!} x^n
\]
Rewrite the power series in summation notation explicitly as:
\[
\displaystyle e^{2x} = \sum_{n=0}^{\infty} \frac{(2x)^n}{n!}
\]
This is the power series representation of \( e^{2x} \) centered at \( a = 0 \). The interval of convergence for this series is all real numbers \( (-\infty, \infty) \) because the exponential function's power series converges everywhere.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Taylor Series Expansion
The Taylor series represents a function as an infinite sum of terms calculated from the derivatives of the function at a single point, called the center (a). For f(x) = e^(2x) at a = 0, the series uses derivatives evaluated at 0 to express f(x) as a power series in (x - 0).
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08:42
Taylor Series
Power Series and Summation Notation
A power series is an infinite sum of terms in the form c_n(x - a)^n, where c_n are coefficients and a is the center. Summation notation compactly expresses this series as Σ c_n (x - a)^n, making it easier to write and analyze the series representation of 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 testing the radius within which the infinite series converges, ensuring the series accurately represents the function within that range.
Recommended video:
08:44Interval of Convergence
Related Practice
Textbook Question
Probability: sudden−death playoff Teams A and B go into suddendeath overtime after playing to a tie. The teams alternate possession of the ball, and the first team to score wins. Assume each team has a 1/6 chance of scoring when it has the ball, and Team A has the ball first.
b. The expected number of rounds (possessions by either team) required for the overtime to end is (1/6) ∑ₖ₌₁∞ k(5/6)ᵏ⁻¹. Evaluate this series.
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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
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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) = ln x, a = 3
42
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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
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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.
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