Textbook Question
Taylor series and interval of convergence
b. Write the power series using summation notation.
f(x) = (1 + x²)⁻¹, a = 0
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 11.1.74b
{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) = ln (1 + x) ≈ x − x²/2
Verified step by step guidance1
Identify the function and its approximation: The function given is \(f(x) = \ln(1 + x)\), and the approximation near zero is \(f(x) \approx x - \frac{x^2}{2}\).
Calculate the approximate value at \(x = 0.2\) by substituting into the approximation formula: \(f(0.2) \approx 0.2 - \frac{(0.2)^2}{2}\).
Understand that the error bound can be estimated using the remainder term from the Taylor series expansion of \(\ln(1+x)\) around \(x=0\).
Recall that the Taylor series for \(\ln(1+x)\) is \(x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots\), so the next term after the approximation is \(\frac{x^3}{3}\), which helps in bounding the error.
Use the Lagrange form of the remainder to bound the error: the error \(R_2(x)\) satisfies \(|R_2(x)| \leq \max_{c \in [0, x]} \left| \frac{f^{(3)}(c)}{3!} x^3 \right|\), where \(f^{(3)}(x)\) is the third derivative of \(\ln(1+x)\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Taylor Series Approximation
A Taylor series approximates a function near a point by a polynomial whose terms involve derivatives of the function at that point. For small values of x near zero, ln(1 + x) can be approximated by the first few terms of its Taylor series, such as x − x²/2, simplifying calculations.
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Taylor Series
Error Bound in Taylor Approximations
The error bound quantifies the maximum difference between the actual function value and its Taylor polynomial approximation. It is often estimated using the remainder term, which depends on higher-order derivatives evaluated within the interval, ensuring the approximation's accuracy is understood.
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Properties of the Natural Logarithm Function
The natural logarithm function ln(1 + x) is defined for x > -1 and is smooth and differentiable near zero. Understanding its behavior and derivatives helps in constructing accurate approximations and estimating errors when x is close to zero.
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Related Practice
Textbook Question
{Use of Tech} Fresnel integrals The theory of optics gives rise to the two Fresnel integrals
S(x) = ∫₀ˣ sin t² dt and C(x) = ∫₀ˣ cos t² dt
b. Expand sin t² and cos t² in a Maclaurin series, and then integrate to find the first four nonzero terms of the Maclaurin series for S and C.
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
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.
Textbook Question
Taylor series and interval of convergence
b. Write the power series using summation notation.
f(x) = e²ˣ, a = 0
Textbook Question
Taylor series
b. Write the power series using summation notation.
f(x) = ln x, a = 3