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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 10.4.74b
10장, 문제 10.4.74b

b. From Example 5, Section 10.2, show that
S = 1 + ∑(from n=1 to ∞) [1 / (n²(n + 1))].

검증된 단계별 안내
1
Recall the series expression from Example 5, Section 10.2, which involves the sum of terms related to \( \frac{1}{n^2 (n+1)} \). Our goal is to express \( S \) as \( 1 + \sum_{n=1}^\infty \frac{1}{n^2 (n+1)} \).
Start by writing the general term of the series: \( \frac{1}{n^2 (n+1)} \). To simplify or analyze this term, consider using partial fraction decomposition to break it into simpler fractions that are easier to sum.
Set up the partial fraction decomposition for \( \frac{1}{n^2 (n+1)} \) as \( \frac{A}{n} + \frac{B}{n^2} + \frac{C}{n+1} \), and solve for constants \( A, B, C \) by multiplying both sides by \( n^2 (n+1) \) and equating coefficients.
Once the partial fractions are found, rewrite the sum \( \sum_{n=1}^\infty \frac{1}{n^2 (n+1)} \) as the sum of simpler series involving \( \sum \frac{1}{n} \), \( \sum \frac{1}{n^2} \), and \( \sum \frac{1}{n+1} \).
Recognize that the term \( 1 \) outside the summation corresponds to the initial term or a boundary condition from the original series, completing the expression \( S = 1 + \sum_{n=1}^\infty \frac{1}{n^2 (n+1)} \).

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주요 개념

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

Infinite Series and Summation Notation

An infinite series is the sum of infinitely many terms, often expressed using summation notation (∑). Understanding how to interpret and manipulate these sums is essential for evaluating or transforming series expressions.
추천 영상:
가이드 코스
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Convergence of an Infinite Series

Partial Fraction Decomposition

Partial fraction decomposition breaks a complex rational expression into simpler fractions. This technique is useful for simplifying terms like 1/(n²(n+1)) to facilitate summation or integration.
추천 영상:
가이드 코스
10:07
Partial Fraction Decomposition: Distinct Linear Factors

Telescoping Series

A telescoping series is one where many terms cancel out when the series is expanded, leaving only a few terms to sum. Recognizing and applying telescoping can simplify the evaluation of series like the one given.
추천 영상:
가이드 코스
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Geometric Series
관련 실천
교과서 질문

∑ (from n=1 to ∞) (1 / √(n + 1)) diverges

b. What should n be in order that the partial sum sₙ = ∑ (from i=1 to n) (1 / √(i + 1)) satisfies sₙ > 1000?

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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

Intervals of Convergence

In Exercises 1–36, for what values of x does the series converge (b) absolutely?

∑ (from n = 1 to ∞) [ (√(n + 1) − √n)(x − 3)ⁿ ]

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교과서 질문

Use the Cauchy condensation test from Exercise 59 to show that:

b. ∑ (from n=1 to ∞) [1 / nᵖ] converges if p > 1 and diverges if p ≤ 1.

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교과서 질문

Intervals of Convergence

Intervals of Convergence

In Exercises 1–36, for what values of x does the series converge (b) absolutely?

∑ (from n = 1 to ∞) [ (3x + 1)^(n + 1) / (2n + 2) ]

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교과서 질문

Intervals of Convergence

In Exercises 1–36, for what values of x does the series converge (b) absolutely?

∑ (from n = 0 to ∞) [ (−2)ⁿ (n + 1) (x − 1)ⁿ ]

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