교과서 질문
23–38. Divergence, Integral, and p-series Tests Use the Divergence Test, the Integral Test, or the p-series test to determine whether the following series converge.
∑ (k = 1 to ∞) k^(–1/5)
66
views
Ch. 10 - Sequences and Infinite Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
당신이 사용하는 게 아니라요?교과서 변경다른 교과서를 선택하세요
TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
10장, 문제 10.5.41
40–62. Choose your test Use the test of your choice to determine whether the following series converge.
∑ (k = 1 to ∞) (1 + 2 / k)ᵏ
검증된 단계별 안내1
Identify the given series: \( \sum_{k=1}^{\infty} \left(1 + \frac{2}{k}\right)^k \). This is an infinite series where each term is \( a_k = \left(1 + \frac{2}{k}\right)^k \).
Recognize that the terms \( a_k \) resemble the form \( \left(1 + \frac{x}{k}\right)^k \), which is related to the expression for \( e^x \) as \( k \to \infty \). Here, \( x = 2 \).
Recall the Test for Divergence (also called the nth-term test): if \( \lim_{k \to \infty} a_k \neq 0 \), then the series \( \sum a_k \) diverges. So, compute \( \lim_{k \to \infty} \left(1 + \frac{2}{k}\right)^k \).
Evaluate the limit \( \lim_{k \to \infty} \left(1 + \frac{2}{k}\right)^k = e^2 \), which is a positive finite number greater than zero.
Since the limit of the terms \( a_k \) is not zero, by the Test for Divergence, the series \( \sum_{k=1}^{\infty} \left(1 + \frac{2}{k}\right)^k \) diverges.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
4m주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Convergence of Infinite Series
An infinite series converges if the sum of its terms approaches a finite limit as the number of terms increases indefinitely. Determining convergence involves analyzing the behavior of the terms and applying appropriate tests to see if the series sums to a finite value.
추천 영상:
가이드 코스
06:52
Convergence of an Infinite Series
Root Test
The Root Test evaluates the limit of the k-th root of the absolute value of the terms in a series. 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 k-th power.
추천 영상:
가이드 코스
07:15Root Test
Limit Comparison and Growth of Terms
Understanding how the terms (1 + 2/k)^k behave as k approaches infinity is crucial. This expression resembles the form (1 + 1/n)^n, which approaches e, but with a different coefficient. Comparing the limit of terms to known limits helps determine if terms approach zero, a necessary condition for convergence.
추천 영상:
가이드 코스
07:45Limit Comparison Test
관련 실천
교과서 질문
9–15. Geometric sums Evaluate each geometric sum.
{Use of Tech}∑ k = 0 to 20(2/5)²ᵏ
76
views
교과서 질문
Growth rates of sequences
Use Theorem 10.6 to find the limit of the following sequences or state that they diverge.
aₙ = (6ⁿ + 3ⁿ) / (6ⁿ + n¹⁰⁰)
70
views
교과서 질문
72–86. Evaluating series Evaluate each series or state that it diverges.
∑ (k = 0 to ∞) (1/4)ᵏ × 5^(3 – k)
42
views
교과서 질문
21–42. Geometric series Evaluate each geometric series or state that it diverges.
35.∑ (k = 0 to ∞) 3(–π)^(–k)
74
views
교과서 질문
45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (k = 2 to ∞) (−1)ᵏ · k · (k² + 1) / (k³ − 1)
65
views