교과서 질문
{Use of Tech} 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.
∫₀⁰ᐧ³⁵ tan ⁻¹x dx
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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 11.2.32
Radius and interval of convergence Determine the radius and interval of convergence of the following power series.
∑ₖ₌₀∞ (-x/10)²ᵏ
검증된 단계별 안내1
Rewrite the given power series in a more standard form. The series is \( \sum_{k=0}^\infty \left(-\frac{x}{10}\right)^{2k} \). Notice that \( \left(-\frac{x}{10}\right)^{2k} = \left(\frac{x^2}{100}\right)^k \). So the series can be expressed as \( \sum_{k=0}^\infty \left(\frac{x^2}{100}\right)^k \).
Recognize that this is a geometric series with common ratio \( r = \frac{x^2}{100} \). A geometric series \( \sum_{k=0}^\infty r^k \) converges if and only if \( |r| < 1 \).
Set up the inequality for convergence: \( \left| \frac{x^2}{100} \right| < 1 \). Since \( x^2 \geq 0 \), this simplifies to \( \frac{x^2}{100} < 1 \).
Solve the inequality for \( x \): multiply both sides by 100 to get \( x^2 < 100 \), then take the square root to find \( |x| < 10 \). This gives the radius of convergence \( R = 10 \).
Determine the interval of convergence by considering the endpoints \( x = -10 \) and \( x = 10 \). Substitute these values back into the original series and test for convergence at each endpoint to finalize the interval.
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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 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 behaves well.
추천 영상:
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 they are included.
추천 영상:
08:44Interval of Convergence
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