Textbook Question
Approximating ln 2 Consider the following three ways to approximate
ln 2.
e. Using four terms of the series, which of the three series derived in parts (a)–(d) gives the best approximation to ln 2? Can you explain why?
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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.RE.14
Find the remainder term Rₙ(x) for the Taylor series centered at 0 for the following functions. Find an upper bound for the magnitude of the remainder on the given interval for the given value of n. (The bound is not unique.)
ƒ(x) = eˣ; bound R₃(x), for |x| < 1
Verified step by step guidance1
Identify the function and the center of the Taylor series: here, \( f(x) = e^x \) and the series is centered at 0 (Maclaurin series).
Recall the Taylor series expansion of \( e^x \) at 0: \( e^x = \sum_{k=0}^\infty \frac{x^k}{k!} \). The remainder term after \( n \) terms is given by the Lagrange form of the remainder:
\[ R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!} x^{n+1} \] where \( c \) is some value between 0 and \( x \).
Since all derivatives of \( e^x \) are \( e^x \), we have \( f^{(n+1)}(c) = e^c \). To find an upper bound for \( |R_3(x)| \) on \( |x| < 1 \), note that \( |c| < 1 \) and \( e^c \) is increasing on \( \mathbb{R} \). So the maximum value of \( e^c \) on \( |c| < 1 \) is \( e^1 = e \).
Therefore, the upper bound for the magnitude of the remainder term is:
\[ |R_3(x)| \leq \frac{e}{4!} |x|^4 \] for all \( |x| < 1 \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Taylor Series and Remainder Term
A Taylor series represents a function as an infinite sum of terms calculated from the function's derivatives at a single point. The remainder term Rₙ(x) measures the error between the function and its nth-degree Taylor polynomial, indicating how closely the polynomial approximates the function near the center.
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Taylor Series
Lagrange Form of the Remainder
The Lagrange form expresses the remainder term Rₙ(x) as Rₙ(x) = (f⁽ⁿ⁺¹⁾(ξ) / (n+1)!) * xⁿ⁺¹ for some ξ between 0 and x. This form helps estimate the error by bounding the (n+1)th derivative on the interval, providing a practical way to find an upper bound for the remainder.
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Bounding the Remainder on an Interval
To find an upper bound for |Rₙ(x)| on an interval, identify the maximum absolute value of the (n+1)th derivative within that interval. For eˣ, all derivatives are eˣ, which is maximized at the endpoint of the interval. Multiplying this maximum by |x|ⁿ⁺¹/(n+1)! gives a valid upper bound for the remainder.
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Related Practice
Textbook Question
Taylor polynomials Find the nth-order Taylor polynomial for the following functions centered at the given point a.
ƒ(x) = e^(sin x), n = 2, a = 0
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Textbook Question
Approximating ln 2 Consider the following three ways to approximate
ln 2.
b. Use the Taylor series for ln (1 - x) centered at 0 and the identity ln 2 = -ln 1/2. Write the resulting infinite series.
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Textbook Question
Radius and interval of convergence Use the Ratio Test or the Root Test to determine the radius of convergence of the following power series. Test the endpoints to determine the interval of convergence, when appropriate.
∞
Σ x⁴ᵏ/k²
k = 1
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Textbook Question
ƒ(x) = eˣ, a = 0; e-0.08
b. Use the Taylor polynomials to approximate the given expression. Make a table showing the approximations and the absolute error in these approximations using a calculator for the exact function value.
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Textbook Question
Radius and interval of convergence Use the Ratio Test or the Root Test to determine the radius of convergence of the following power series. Test the endpoints to determine the interval of convergence, when appropriate.
x +x³/3 +x⁵/5 +x⁷/7 + ...
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