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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 10.7.6
10장, 문제 10.7.6

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 = 0 to ∞) (2x)ⁿ

검증된 단계별 안내
1
Identify the given series: \( \sum_{n=0}^{\infty} (2x)^n \). This is a geometric series with common ratio \( r = 2x \).
Recall that a geometric series \( \sum r^n \) converges if and only if \( |r| < 1 \). So, set up the inequality \( |2x| < 1 \) to find the interval of convergence.
Solve the inequality \( |2x| < 1 \) to get \( |x| < \frac{1}{2} \). This gives the radius of convergence \( R = \frac{1}{2} \) and the open interval \( (-\frac{1}{2}, \frac{1}{2}) \).
Check the endpoints \( x = -\frac{1}{2} \) and \( x = \frac{1}{2} \) by substituting into the series to determine if the series converges at these points. Since the series becomes \( \sum (-1)^n \) or \( \sum 1^n \), analyze their convergence behavior.
Determine absolute convergence by checking if the series \( \sum |(2x)^n| = \sum (2|x|)^n \) converges, which happens when \( |x| < \frac{1}{2} \). For conditional convergence, check if the series converges at endpoints where absolute convergence fails.

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주요 개념

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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. Finding these involves applying tests like the Ratio Test to determine where the series converges absolutely or conditionally.
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, ensuring the original series converges regardless of term signs. Absolute convergence guarantees stronger convergence properties and often simplifies analysis of power series.
추천 영상:
가이드 코스
07:51
Choosing a Convergence Test

Conditional Convergence

Conditional convergence occurs when a series converges, but not absolutely; that is, the series ∑a_n converges while ∑|a_n| diverges. This typically happens at the endpoints of the interval of convergence and requires careful testing, such as the Alternating Series Test, to determine convergence behavior.
추천 영상:
가이드 코스
07:51
Choosing a Convergence Test
관련 실천
교과서 질문

Limit Comparison Test

In Exercises 9–16, use the Limit Comparison Test to determine if each series converges or diverges.

∑ (from n=2 to ∞) 1 / ln n

(Hint: Limit Comparison with ∑ (from n=2 to ∞) (1/n))

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교과서 질문

Applying the Integral Test

Use the Integral Test to determine if the series in Exercises 1–12 converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied.

∑ (from n = 2 to ∞) ln(n²) / n

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교과서 질문

Telescoping Series

In Exercises 39–44, find a formula for the nth partial sum of the series and use it to determine if the series converges or diverges. If a series converges, find its sum.

∑ (from n = 1 to ∞) [ (1/n) − (1/(n + 1)) ]

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교과서 질문

Are there any values of x for which ∑ (from n=1 to ∞) (1 / nˣ) converges? Give reasons for your answer.

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교과서 질문

Determining Convergence or Divergence

In Exercises 1–14, determine whether the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test.

∑ (from n = 2 to ∞) [(-1)ⁿ⁺¹ (1 / ln n)]

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교과서 질문

Using the Ratio Test

In Exercises 1–8, use the Ratio Test to determine whether each series converges absolutely or diverges.

∑(from n=1 to ∞) [(-1)ⁿ (n + 2) / 3ⁿ]

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