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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.AAE.30b
10장, 문제 10.AAE.30b

30. b. By differentiating the series in part (a) term by term, show that
Σ(from n=1 to ∞) n / (n + 1)! = 1.

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Recall the series from part (a), which is likely the sum of the reciprocals of factorials, such as \( \sum_{n=0}^\infty \frac{1}{n!} = e \). Identify the exact series given in part (a) to differentiate term by term.
Express the series \( \sum_{n=1}^\infty \frac{n}{(n+1)!} \) in a form that relates to the series from part (a). Notice that \( \frac{n}{(n+1)!} = \frac{n+1-1}{(n+1)!} = \frac{n+1}{(n+1)!} - \frac{1}{(n+1)!} = \frac{1}{n!} - \frac{1}{(n+1)!} \).
Rewrite the series using this decomposition: \( \sum_{n=1}^\infty \frac{n}{(n+1)!} = \sum_{n=1}^\infty \left( \frac{1}{n!} - \frac{1}{(n+1)!} \right) \).
Recognize that this is a telescoping series, where most terms cancel out when expanded. Write out the first few terms explicitly to see the cancellation pattern.
Sum the telescoping series by taking the limit as \( n \to \infty \) of the partial sums, and conclude that the sum converges to 1.

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주요 개념

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

Term-by-Term Differentiation of Series

Term-by-term differentiation allows differentiating an infinite series by differentiating each term individually, provided the series converges uniformly. This technique is useful to find new series representations or to simplify expressions involving series.
추천 영상:
가이드 코스
05:44
Divergence Test (nth Term Test)

Factorials and Series Manipulation

Factorials (n!) grow rapidly and often appear in series expansions like those of exponential functions. Understanding how to manipulate terms involving factorials, such as rewriting n/(n+1)! as n!/(n+1)!, is key to simplifying and evaluating series.
추천 영상:
5:22
Factorials

Convergence of Infinite Series

Determining whether an infinite series converges is essential before performing operations like differentiation term-by-term. Recognizing convergence ensures the validity of the manipulations and that the series sums to a finite value, such as the sum equaling 1 in this problem.
추천 영상:
가이드 코스
06:52
Convergence of an Infinite Series
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교과서 질문

Theory and Examples

Suppose that a₁, a₂, a₃, …, aₙ are positive numbers satisfying the following conditions:

i) a₁ ≥ a₂ ≥ a₃ ≥ …;


ii) the series a₂ + a₄ + a₈ + a₁₆ + … diverges.

Show that the series


a₁/1 + a₂/2 + a₃/3 + …


diverges.

19
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교과서 질문

Power Series

In Exercises 47–56, (a) find the series’ radius and interval of convergence. Then identify the values of x for which the series converges (b) absolutely and (c) conditionally.


∑ (from n = 1 to ∞) (x + 4)ⁿ/(n3ⁿ)

34
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교과서 질문

Absolute and Conditional Convergence

Which of the series in Exercises 15–48 converge absolutely, which converge, and which diverge? Give reasons for your answers.

∑ (from n = 1 to ∞) [(-1)ⁿ (√(n² + n) − n)]

32
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교과서 질문

Find the sum of each series in Exercises 45–52.

∑ (from n = 1 to ∞) [ (2n + 1) / (n²(n + 1)²) ]

20
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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.

∑ (from n = 2 to ∞) [(ln n / n)³]

6
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교과서 질문

Determining Convergence of Series

Which of the series in Exercises 25–44 converge absolutely, which converge conditionally, and which diverge? Give reasons for your answers.


∑ (from n = 2 to ∞) 1/[n(ln n)²]