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Ch. 11 - Power Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Briggs 3rd EditionCh. 11 - Power Series문제 11.2.30
11장, 문제 11.2.30

Radius and interval of convergence Determine the radius and interval of convergence of the following power series.
∑ₖ₌₀∞ k(x−1)ᵏ

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Identify the given power series: \(\sum_{k=0}^{\infty} k (x - 1)^k\). This is a power series centered at \(x = 1\).
To find the radius of convergence, use the Ratio Test. Consider the general term \(a_k = k (x - 1)^k\). Compute the limit \(L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right|\).
Calculate the ratio inside the limit: \(\left| \frac{(k+1)(x-1)^{k+1}}{k (x-1)^k} \right| = \left| \frac{k+1}{k} \right| \cdot |x-1| = \left(1 + \frac{1}{k}\right) |x-1|\).
Take the limit as \(k \to \infty\): \(L = \lim_{k \to \infty} \left(1 + \frac{1}{k}\right) |x-1| = |x-1|\). The Ratio Test says the series converges if \(L < 1\), so the radius of convergence \(R\) satisfies \(|x-1| < 1\).
Determine the interval of convergence by checking the endpoints \(x = 1 - 1 = 0\) and \(x = 1 + 1 = 2\). Substitute these values into the series and test for convergence using appropriate tests (e.g., p-series, alternating series, or divergence test).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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 the structure of power series is essential to analyze their convergence behavior around the center point.
추천 영상:
05:58
Intro to Power Series

Radius of Convergence

The radius of convergence is the distance from the center within which the power series converges absolutely. It can be found using tests like the Ratio Test or Root Test, and it defines the interval where the series behaves well.
추천 영상:
07:36
Radius of Convergence

Interval of Convergence

The interval of convergence includes all x-values for which the power series converges, typically centered at c and extending radius R in both directions. Endpoints must be checked separately to determine if the series converges or diverges there.
추천 영상:
08:44
Interval of Convergence
관련 실천
교과서 질문

Remainders Find the remainder Rₙ for the nth−order Taylor polynomial centered at a for the given functions. Express the result for a general value of n.


f(x) = sin x, a = π/2

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교과서 질문

Derivative trick Here is an alternative way to evaluate higher derivatives of a function f that may save time. Suppose you can find the Taylor series for f centered at the point a without evaluating derivatives (for example, from a known series). Then f⁽ᵏ⁾(a)=k! multiplied by the coefficient of (x−a)ᵏ. Use this idea to evaluate f⁽³⁾(0) and f⁽⁴⁾(0) for the following functions. Use known series and do not evaluate derivatives.


f(x) = ∫₀ˣ sin t² dt

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교과서 질문

Representing functions by power series Identify the functions represented by the following power series.

∑ₖ₌₀∞ 2ᵏ x²ᵏ⁺¹

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교과서 질문

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) = eˣ

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교과서 질문

Does the accuracy of an approximation given by a Taylor polynomial generally increase or decrease with the order of the approximation? Explain.

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교과서 질문

Limits Evaluate the following limits using Taylor series.

lim ₓ→₄ (x² 16)/(ln (x 3)}

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