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Ch. 11 - Power Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 11 - Power SeriesProblem 11.2.23
Chapter 11, Problem 11.2.23

Radius and interval of convergence Determine the radius and interval of convergence of the following power series.


∑ₖ₌₁∞ ((−1)ᵏ⁺¹(x−1)ᵏ)/k

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1
Identify the given power series: \( \sum_{k=1}^{\infty} \frac{(-1)^{k+1} (x-1)^k}{k} \). This is a power series centered at \( x = 1 \).
To find the radius of convergence, use the Ratio Test or Root Test. Here, the Ratio Test is convenient. Consider the general term \( a_k = \frac{(-1)^{k+1} (x-1)^k}{k} \).
Apply the Ratio Test by computing \( L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| = \lim_{k \to \infty} \left| \frac{(x-1)^{k+1}}{k+1} \cdot \frac{k}{(x-1)^k} \right| = \lim_{k \to \infty} |x-1| \cdot \frac{k}{k+1} = |x-1| \).
The Ratio Test states the series converges if \( L < 1 \), so the radius of convergence \( R = 1 \). This means the series converges for \( |x-1| < 1 \), or \( x \in (0, 2) \).
Check the endpoints \( x=0 \) and \( x=2 \) by substituting into the series and testing for convergence (e.g., using the Alternating Series Test or p-series test) to determine the interval of convergence.

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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 is essential for studying convergence behavior around the center point.
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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 or Root Test, and it defines the interval where the series represents a valid function.
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Radius of Convergence

Interval of Convergence

The interval of convergence includes all x-values for which the power series converges, possibly including endpoints. After finding the radius, endpoint testing is necessary to determine if the series converges or diverges at those boundary points.
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Interval of Convergence
Related Practice
Textbook Question

Combining power series Use the geometric series


f(x) = 1/(1-x) = ∑ₖ₌₀∞ xᵏ, for |x| < 1,


to find the power series representation for the following functions (centered at 0). Give the interval of convergence of the new series.


f(x³) = 1/(1 − x³)

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Textbook Question

Suppose you use a second-order Taylor polynomial centered at 0 to approximate a function f. What matching conditions are satisfied by the polynomial?

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Textbook Question

Working with binomial series Use properties of power series, substitution, and factoring to find the first four nonzero terms of the Maclaurin series for the following functions. Use the Maclaurin series


(1 + x)⁻² = 1 − 2x + 3x² − 4x³ + ⋯, for −1 < x < 1.


1/(3 + 4x)²

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Textbook Question

Tangent line is p₁ Let f be differentiable at x=a


a. Find the equation of the line tangent to the curve y=f(x) at (a, f(a)).


b. Verify that the Taylor polynomial p_1 centered at a describes the tangent line found in part (a).

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Textbook Question

Power series for derivatives


a. Differentiate the Taylor series centered at 0 for the following functions.

b. Identify the function represented by the differentiated series.

c. Give the interval of convergence of the power series for the derivative.


f(x) = ln (1 + x)

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Textbook Question

Approximating definite integrals Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than 10⁻⁴.∫₀⁰ᐧ²⁵ e⁻ˣ² dx

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