교과서 질문
Assume that the series ∑ aₙ(x − 2)ⁿ converges for x = −1 and diverges for x = 6. Answer true (T), false (F), or not enough information given (N) for the following statements about the series.
f. Diverges for x = 4.9
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Ch. 10 - Infinite Sequences and Series
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
10장, 문제 10.7.28c
Intervals of Convergence
In Exercises 1–36, for what values of x does the series converge (c) conditionally?
∑ (from n = 0 to ∞) [ (−2)ⁿ (n + 1) (x − 1)ⁿ ]
검증된 단계별 안내1
Identify the given power series: \(\sum_{n=0}^{\infty} (-2)^n (n+1) (x-1)^n\).
Rewrite the series in a form that isolates the variable term: \(\sum_{n=0}^{\infty} (n+1) \left[-2(x-1)\right]^n\).
Use the Root Test or Ratio Test to find the radius of convergence. For the Ratio Test, consider the limit \(L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\), where \(a_n = (n+1) \left[-2(x-1)\right]^n\).
Solve the inequality \(L < 1\) to find the interval of convergence in terms of \(x\).
Check the endpoints of the interval separately by substituting them back into the original series to determine if the series converges absolutely, conditionally, or diverges at those points.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
7m주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Radius and Interval of Convergence
The radius of convergence determines the distance from the center point within which a power series converges absolutely. The interval of convergence includes all x-values for which the series converges, possibly including endpoints where convergence must be tested separately.
추천 영상:
07:36
Radius of Convergence
Absolute vs. Conditional Convergence
A series converges absolutely if the series of absolute values converges; otherwise, it may converge conditionally if the original series converges but not absolutely. Conditional convergence often occurs at the endpoints of the interval of convergence.
추천 영상:
가이드 코스
07:51Choosing a Convergence Test
Ratio Test for Convergence
The ratio test uses the limit of the absolute value of the ratio of consecutive terms to determine convergence. If this limit is less than one, the series converges absolutely; if greater than one, it diverges; if equal to one, the test is inconclusive and further analysis is needed.
추천 영상:
가이드 코스
05:35Ratio Test
관련 실천
교과서 질문
Intervals of Convergence
In Exercises 1–36, for what values of x does the series converge (c) conditionally?
∑ (from n = 1 to ∞) [ (3x + 1)^(n + 1) / (2n + 2) ]
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교과서 질문
A sequence of rational numbers is described as follows:
1/1,3/2,7/5,17/12,…,a/b,(a + 2b)/(a + b),…
Here the numerators form one sequence, the denominators form a second sequence, and their ratios form a third sequence. Let xₙ and yₙ be, respectively, the numerator and the denominator of the nᵗʰ fraction rₙ = xₙ / yₙ.
b. The fractions rₙ = xₙ / yₙ approach a limit as n increases. What is that limit? (Hint: Use part (a) to show that rₙ² − 2 = ±(1 / yₙ)² and that yₙ is not less than n.)
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교과서 질문
Assume that the series ∑ aₙxⁿ converges for x = 4 and diverges for x = 7. Answer true (T), false (F), or not enough information given (N) for the following statements about the series.
e. Diverges for x = 8
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교과서 질문
Intervals of Convergence
In Exercises 1–36, for what values of x does the series converge (c) conditionally?
∑ (from n = 1 to ∞) [ (√(n + 1) − √n)(x − 3)ⁿ ]
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