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?
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 11.2.9
Radius and interval of convergence Determine the radius and interval of convergence of the following power series.
∑ₖ₌₀∞ (2x)ᵏ
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Identify the given power series: \(\sum_{k=0}^{\infty} (2x)^k\). This is a geometric series with the general term \(a_k = (2x)^k\).
Recall that a geometric series \(\sum_{k=0}^{\infty} r^k\) converges if and only if \(|r| < 1\). Here, the ratio \(r\) is \$2x$.
Set up the inequality for convergence: \(|2x| < 1\). This inequality will help us find the radius of convergence.
Solve the inequality \(|2x| < 1\) to find the interval for \(x\): dividing both sides by 2 gives \(|x| < \frac{1}{2}\). This means the radius of convergence \(R\) is \(\frac{1}{2}\).
Determine the interval of convergence by considering the endpoints \(x = -\frac{1}{2}\) and \(x = \frac{1}{2}\). Test these values in the original series to check if the series converges or diverges at the endpoints.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Power Series
A power series is an infinite sum of terms in the form a_k(x - c)^k, where a_k are coefficients and c is the center. Understanding power series helps analyze functions as infinite polynomials and study their convergence behavior around the center point.
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05:58
Intro to Power Series
Radius of Convergence
The radius of convergence is the distance from the center within which a power series converges absolutely. It can be found using tests like the Ratio Test, and it defines the interval where the series represents a valid function.
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07:36Radius of Convergence
Interval of Convergence
The interval of convergence is the set of x-values for which the power series converges. It includes all points within the radius of convergence and requires checking endpoints separately to determine if the series converges there.
Recommended video:
08:44Interval of Convergence
Related Practice
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.
sin 0.3, n = 4
Textbook Question
Radius and interval of convergence Determine the radius and interval of convergence of the following power series.
∑ₖ₌₂∞ ((x+3)ᵏ)/(k łn²k)
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Textbook Question
Any method
a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. You do not need to use the definition of the Taylor series coefficients.
b. Determine the radius of convergence of the series.
f(x) = (1 + x²)⁻²/³
Textbook Question
Manipulating Taylor series Use the Taylor series in Table 11.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0.
ln (1 + x²)