교과서 질문
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.
(1 + x⁴)⁻¹
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Ch. 11 - Power Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- 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
11장, 문제 11.2.11
Radius and interval of convergence Determine the radius and interval of convergence of the following power series.
∑ₖ₌₁∞ (kx)ᵏ
검증된 단계별 안내1
Identify the given power series: \( \sum_{k=1}^{\infty} (kx)^k \). This can be rewritten as \( \sum_{k=1}^{\infty} k^k x^k \).
To find the radius of convergence, use the root test which involves the limit \( L = \lim_{k \to \infty} \sqrt[k]{|a_k|} \), where \( a_k = k^k x^k \).
Calculate \( \sqrt[k]{|a_k|} = \sqrt[k]{k^k |x|^k} = k |x| \).
Find the limit \( L = \lim_{k \to \infty} k |x| \). For the series to converge, this limit must be less than 1, so set \( \lim_{k \to \infty} k |x| < 1 \).
Since \( \lim_{k \to \infty} k |x| = \infty \) for any \( x \neq 0 \), the radius of convergence is 0, meaning the series converges only at \( x = 0 \). The interval of convergence is therefore \( \{0\} \).
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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
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4m주요 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 depending on the variable x.
추천 영상:
05:58
Intro to Power Series
Radius of Convergence
The radius of convergence is the distance from the center c 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:
07:36Radius of Convergence
Interval of Convergence
The interval of convergence is the set of all 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.
추천 영상:
08:44Interval of Convergence
관련 실천
교과서 질문
Functions to power series Find power series representations centered at 0 for the following functions using known power series. Give the interval of convergence for the resulting series.
f(x) = ln √(4 − x²)
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교과서 질문
Shifting power series If the power series f(x)=∑ cₖ xᵏ has an interval of convergence of |x|<R, what is the interval of convergence of the power series for f(x−a), where a ≠ 0 is a real number?
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교과서 질문
{Use of Tech} Best center point Suppose you wish to approximate cos (π/ 2) using Taylor polynomials. Is the approximation more accurate if you use Taylor polynomials centered at 0 or at π/6? Use a calculator for numerical experiments and check for consistency with Theorem 11.2. Does the answer depend on the order of the polynomial?
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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) = (1 − x)⁻¹
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교과서 질문
Suppose f(0)=1, f'(0)=0, f''(0)=2, and f⁽³⁾(0)=6. Find the third-order Taylor polynomial for f centered at 0 and use it to approximate f(0.2).
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