Textbook Question
Radius and interval of convergence Determine the radius and interval of convergence of the following power series.
∑ₖ₌₀∞ (2x)ᵏ
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 11.1.47
{Use of Tech} Estimating errors Use the remainder to find a bound on the error in approximating the following quantities with the nth-order Taylor polynomial centered at 0. Estimates are not unique.
sin 0.3, n = 4
Verified step by step guidance1
Identify the function and the point of expansion: here, the function is \(f(x) = \sin x\), and the Taylor polynomial is centered at \(0\) (Maclaurin series).
Recall the Taylor series for \(\sin x\) centered at \(0\):
\(\sin x = \sum_{k=0}^{\infty} (-1)^k \frac{x^{2k+1}}{(2k+1)!}\).
The 4th-order Taylor polynomial means including terms up to degree 4, which for \(\sin x\) corresponds to terms up to \(x^3\) because the series has only odd powers.
Determine the remainder term (Lagrange form) for the Taylor polynomial of order \(n=4\) centered at 0:
\(R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!} x^{n+1}\),
where \(c\) is some value between \(0\) and \(x\).
Since \(n=4\), the remainder involves the 5th derivative of \(\sin x\). Find \(f^{(5)}(x)\) and note its maximum absolute value on the interval between \(0\) and \(0.3\) to bound the error.
Use the bound on \(|f^{(5)}(c)|\) and plug into the remainder formula:
\(|R_4(0.3)| \leq \frac{M}{5!} |0.3|^5\),
where \(M\) is the maximum of \(|f^{(5)}(c)|\) for \(c\) in \([0, 0.3]\). This gives a bound on the error in approximating \(\sin 0.3\) by the 4th-order Taylor polynomial.
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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 a finite sum of its derivatives at that point. The nth-order Taylor polynomial centered at 0 (Maclaurin polynomial) uses derivatives up to order n to estimate the function's value close to zero.
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Taylor Polynomials
Remainder (Error) Term in Taylor Series
The remainder term quantifies the difference between the actual function value and its Taylor polynomial approximation. It provides an upper bound on the error, ensuring the approximation's accuracy within a specified range.
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08:42Taylor Series
Bounding the Error for sin(x)
For sin(x), the remainder after the nth-order polynomial can be bounded using the next derivative's maximum value on the interval. Since derivatives of sin(x) are bounded by 1, this helps estimate the maximum possible error when approximating sin(0.3) with a 4th-order polynomial.
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04:57Determining Error and Relative Error
Related Practice
Textbook Question
Suppose you know the Maclaurin series for f and that it converges to f(x) for |x|<1. How do you find the Maclaurin series for f(x²) and where does it converge?
Textbook Question
{Use of Tech} A different kind of approximation When approximating a function f using a Taylor polynomial, we use information about f and its derivative at one point. An alternative approach (called interpolation) uses information about f at several different points. Suppose we wish to approximate f(x)=sin x on the interval [0, π].
a. Write the (quadratic) Taylor polynomial p₂ for f centered at π/2.
b. Now consider a quadratic interpolating polynomial q(x) = ax² + bx + c. The coefficients a, b, and c are chosen such that the following conditions are satisfied:
q(0) = f(0), q(π/2) = f(π/2), and q(π) = f(π)
Show that q(x) = −(4/π²)x² + (4/π)x.
c. Graph f, p₂, and q on [0, π].
d. Find the error in approximating f(x) = sin x at the points π/4, π/2, 3π/4, and π using p₂ and q.
e. Which function, p₂ or q, is a better approximation to f on [0, π]? Explain.
Textbook Question
{Use of Tech} Estimating errors Use the remainder to find a bound on the error in approximating the following quantities with the nth-order Taylor polynomial centered at 0. Estimates are not unique.
ln 1.04, n=3
Textbook Question
{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.
√(1+x) ≈ 1 + x/2 on [−0.1,0.1]
Textbook Question
Probability: tossing for a head The expected (average) number of tosses of a fair coin required to obtain the first head is ∑ₖ₌₁∞ k(1/2)ᵏ. Evaluate this series and determine the expected number of tosses. (Hint: Differentiate a geometric series.)