Textbook Question
A useful substitution Replace x with x−1 in the series ln (1+x) = ∑ₖ₌₁∞ ((−1)ᵏ⁺¹ xᵏ)/k to obtain a power series for ln x centered at x = 1. What is the interval of convergence for the new power series?
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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 11.1.56
{Use of Tech} Maximum error Use the remainder term to find a bound on the error in the following approximations on the given interval. Error bounds are not unique.
tan x ≈ x on [−π/6, π/6]
Verified step by step guidance1
Identify the function and the approximation: here, the function is \(f(x) = \tan x\) and the approximation is \(f(x) \approx x\) on the interval \(\left[-\frac{\pi}{6}, \frac{\pi}{6}\right]\).
Recognize that the approximation \(\tan x \approx x\) is the first-degree Taylor polynomial of \(\tan x\) centered at 0, so the error can be estimated using the Lagrange remainder term for the Taylor series expansion.
Recall the Lagrange remainder formula for the Taylor polynomial of degree 1 centered at 0: the error \(R_1(x)\) satisfies
\[R_1(x) = \frac{f^{(2)}(c)}{2!} x^2\]
for some \(c\) between 0 and \(x\).
Calculate the second derivative of \(f(x) = \tan x\). First, \(f'(x) = \sec^2 x\), then
\[f^{(2)}(x) = \frac{d}{dx} (\sec^2 x) = 2 \sec^2 x \tan x.\]
To find the error bound, determine the maximum value of \(|f^{(2)}(c)|\) on the interval \(\left[-\frac{\pi}{6}, \frac{\pi}{6}\right]\), then use it in the remainder formula:
\[|R_1(x)| \leq \frac{\max_{c \in [-\frac{\pi}{6}, \frac{\pi}{6}]} |f^{(2)}(c)|}{2} x^2.\]
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Taylor Polynomial Approximation
A Taylor polynomial approximates a function near a point using derivatives at that point. For tan x near 0, the linear approximation is tan x ≈ x, which is the first-degree Taylor polynomial. Understanding this helps in estimating how close the approximation is to the actual function.
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Taylor Polynomials
Remainder Term (Lagrange Form)
The remainder term quantifies the error between the function and its Taylor polynomial approximation. The Lagrange form expresses this error using a higher-order derivative evaluated at some point in the interval, providing a bound on the maximum possible error.
Recommended video:
06:32Alternating Series Remainder
Error Bound on an Interval
To find an error bound on [−π/6, π/6], we evaluate the maximum value of the relevant derivative in that interval. This maximum derivative value, combined with the remainder formula, gives a guaranteed upper limit on the approximation error over the entire interval.
Recommended video:
04:57Determining Error and Relative Error
Related Practice
Textbook Question
Exponential function In Section 11.3, we show that the power series for the exponential function centered at 0 is
eˣ = ∑ₖ₌₀∞ (xᵏ)/k!, for −∞ < x < ∞
Use the methods of this section to find the power series centered at 0 for the following functions. Give the interval of convergence for the resulting series.
f(x) = e⁻³ˣ
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Textbook Question
Radius and interval of convergence Determine the radius and interval of convergence of the following power series.
∑ₖ₌₁∞ (3x + 2)ᵏ/k
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Textbook Question
Inverse sine Given the power series
1/√(1 − x²) = 1 + (1/2)x² + (1 ⋅ 3)/(2 ⋅ 4) x⁴ + (1 ⋅ 3 ⋅ 5)/(2 ⋅ 4 ⋅ 6) x⁶ +⋯
for −1<x<1, find the power series for f(x) = sin ⁻¹ x centered at 0.
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Textbook Question
Limits Evaluate the following limits using Taylor series.
lim ₓ→₁ (x 1)/(ln x)
Textbook Question
{Use of Tech} Number of terms What is the minimum order of the Taylor polynomial required to approximate the following quantities with an absolute error no greater than 10⁻³ ? (The answer depends on your choice of a center.)
ln 0.85