교과서 질문
Does the series
∑ (from n=1 to ∞) (1/n − 1/n²)
converge or diverge? Justify your answer.
55
views
Ch. 10 - Infinite Sequences and Series
Hass15th EditionThomas' CalculusISBN: 9780137616077
당신이 사용하는 게 아니라요?교과서 변경다른 교과서를 선택하세요
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 10.4.25
Determining Convergence or Divergence
Which of the series in Exercises 17–56 converge, and which diverge? Use any method, and give reasons for your answers.
∑ (from n=1 to ∞) (n / (3n + 1))ⁿ
검증된 단계별 안내1
Identify the given series: \( \sum_{n=1}^{\infty} \left( \frac{n}{3n + 1} \right)^n \). This is a series with terms raised to the power \(n\), suggesting the Root Test might be effective.
Recall the Root Test: For a series \( \sum a_n \), compute \( L = \lim_{n \to \infty} \sqrt[n]{|a_n|} \). If \( L < 1 \), the series converges absolutely; if \( L > 1 \), it diverges; if \( L = 1 \), the test is inconclusive.
Apply the Root Test to the terms \( a_n = \left( \frac{n}{3n + 1} \right)^n \). Compute \( \sqrt[n]{|a_n|} = \frac{n}{3n + 1} \).
Evaluate the limit \( L = \lim_{n \to \infty} \frac{n}{3n + 1} \). Simplify the expression by dividing numerator and denominator by \(n\) to find \( L = \lim_{n \to \infty} \frac{1}{3 + \frac{1}{n}} \).
Determine the value of \( L \) as \( n \to \infty \), then use the Root Test conclusion to decide if the series converges or diverges based on whether \( L < 1 \), \( L > 1 \), or \( L = 1 \).
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
3m도움이 되었나요?
주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Convergence and Divergence of Infinite Series
An infinite series converges if the sum of its terms approaches a finite limit as the number of terms grows indefinitely. If the sum does not approach a finite value, the series diverges. Understanding this distinction is fundamental to analyzing series behavior.
추천 영상:
가이드 코스
06:52
Convergence of an Infinite Series
Root Test for Series Convergence
The Root Test evaluates the limit of the nth root of the absolute value of the series terms. If this limit is less than 1, the series converges absolutely; if greater than 1, it diverges; if equal to 1, the test is inconclusive. This test is especially useful for series with terms raised to the nth power.
추천 영상:
가이드 코스
07:15Root Test
Behavior of the General Term
Examining the general term's behavior as n approaches infinity helps determine convergence. If the term does not approach zero, the series diverges. For terms like (n/(3n+1))^n, analyzing the base and its limit is crucial to apply convergence tests effectively.
추천 영상:
가이드 코스
05:44Divergence Test (nth Term Test)
관련 실천
교과서 질문
In Exercises 57–82, use any method to determine whether the series converges or diverges. Give reasons for your answer.
∑ (from n = 0 to ∞) [((n + 1) / (n + 2))ⁿ]
5
views
교과서 질문
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 ]
30
views
교과서 질문
Limit Comparison Test
In Exercises 9–16, use the Limit Comparison Test to determine if each series converges or diverges.
∑ (from n=1 to ∞) n(n + 1) / ((n² + 1)(n − 1))
34
views
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
views
교과서 질문
If ∑aₙ is a convergent series of positive terms, prove that ∑sin(aₙ) converges.
30
views