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Ch. 10 - Infinite Sequences and Series
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Hass 15th EditionCh. 10 - Infinite Sequences and Series문제 10.7.10
10장, 문제 10.7.10

Intervals of Convergence
In Exercises 1–36, (a) find the series’ radius and interval of convergence. For what values of x does the series converge (b) absolutely, (c) conditionally?
∑ (from n = 1 to ∞) [ (x − 1)ⁿ / √n ]

검증된 단계별 안내
1
Identify the given power series: \(\sum_{n=1}^{\infty} \frac{(x - 1)^n}{\sqrt{n}}\) and recognize that it is centered at \(x = 1\).
To find the radius of convergence, apply the Root Test or Ratio Test. Here, the Root Test is convenient: consider \(\lim_{n \to \infty} \sqrt[n]{\left| \frac{(x - 1)^n}{\sqrt{n}} \right|} = \lim_{n \to \infty} \frac{|x - 1|}{n^{1/(2n)}}\).
Since \(\lim_{n \to \infty} n^{1/(2n)} = 1\), the limit simplifies to \(|x - 1|\). For convergence, this limit must be less than 1, so the radius of convergence is \(R = 1\) and the interval of convergence is initially \(1 - 1 < x < 1 + 1\), or \((0, 2)\).
Next, check convergence at the endpoints \(x = 0\) and \(x = 2\) by substituting into the series: at \(x=0\), the series becomes \(\sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}\); at \(x=2\), it becomes \(\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}\). Analyze these using the Alternating Series Test and p-series test respectively.
Determine absolute convergence by testing \(\sum_{n=1}^{\infty} \left| \frac{(x - 1)^n}{\sqrt{n}} \right| = \sum_{n=1}^{\infty} \frac{|x - 1|^n}{\sqrt{n}}\). This series converges absolutely when \(|x - 1| < 1\). For conditional convergence, check if the series converges at endpoints where absolute convergence fails.

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영상 길이:
16m

주요 개념

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

Radius and Interval of Convergence

The radius of convergence is the distance from the center of a power series within which the series converges. The interval of convergence includes all x-values for which the series converges, possibly including endpoints. It is found using tests like the Ratio or Root Test applied to the general term.
추천 영상:
07:36
Radius of Convergence

Absolute Convergence

A series converges absolutely if the series of absolute values converges. This means ∑|a_n| converges, which guarantees convergence regardless of term signs. Absolute convergence implies regular convergence and is often easier to test using comparison or root tests.
추천 영상:
가이드 코스
07:51
Choosing a Convergence Test

Conditional Convergence

Conditional convergence occurs when a series converges, but does not converge absolutely. This means the series ∑a_n converges, but ∑|a_n| diverges. It often happens at the endpoints of the interval of convergence and requires tests like the Alternating Series Test.
추천 영상:
가이드 코스
07:51
Choosing a Convergence Test
관련 실천
교과서 질문

Which series in Exercises 53–76 converge, and which diverge? Give reasons for your answers. If a series converges, find its sum.

∑ (from n = 0 to ∞) e^(−2n)

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교과서 질문

Does the series

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교과서 질문

In Exercises 57–82, use any method to determine whether the series converges or diverges. Give reasons for your answer.

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교과서 질문

Determining Convergence or Divergence

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∑ (from n=1 to ∞) (n / (3n + 1))ⁿ

22
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교과서 질문

Find the first four nonzero terms of the Taylor series for the functions in Exercises 1–10.

6. (1 - x/3)^4

49
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교과서 질문

Suppose that aₙ > 0 and limₙ→∞ n²aₙ = 0. Prove that ∑aₙ converges.

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