교과서 질문
In Exercises 57–82, use any method to determine whether the series converges or diverges. Give reasons for your answer.
∑ (from n = 1 to ∞) [3ⁿ / n³]
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Ch. 10 - Infinite Sequences and Series
Hass15th EditionThomas' CalculusISBN: 9780137616077
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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 10.3.8
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
검증된 단계별 안내1
First, identify the function corresponding to the terms of the series: define \( f(x) = \frac{\ln(x^2)}{x} \) for \( x \geq 2 \).
Check the conditions for the Integral Test: verify that \( f(x) \) is continuous, positive, and decreasing on the interval \([2, \infty)\).
Simplify the function \( f(x) \) using logarithm properties: \( \ln(x^2) = 2 \ln(x) \), so \( f(x) = \frac{2 \ln(x)}{x} \).
Set up the improper integral to apply the Integral Test: \( \int_2^{\infty} \frac{2 \ln(x)}{x} \, dx \).
Evaluate the integral (or analyze its behavior) to determine if it converges or diverges, which will tell you whether the series converges or diverges by the Integral Test.
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주요 개념
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Integral Test for Series Convergence
The Integral Test determines the convergence or divergence of an infinite series by comparing it to an improper integral. If the function corresponding to the series terms is positive, continuous, and decreasing for x ≥ N, then the series and the integral either both converge or both diverge.
추천 영상:
07:51
Choosing a Convergence Test
Conditions for Applying the Integral Test
Before using the Integral Test, verify that the function f(x) representing the series terms is positive, continuous, and decreasing on the interval [N, ∞). These conditions ensure the integral comparison is valid and the test can be applied correctly.
추천 영상:
07:25Integral Test
Evaluating the Improper Integral
To apply the Integral Test, evaluate the improper integral of f(x) from N to infinity. This often involves techniques like substitution or integration by parts. The convergence or divergence of this integral directly determines the behavior of the original series.
추천 영상:
11:11Improper Integrals: Infinite Intervals
관련 실천
교과서 질문
Error Estimation
In Exercises 49–52, estimate the magnitude of the error involved in using the sum of the first four terms to approximate the sum of the entire series.
1 / (1 + t) = ∑ (from n = 0 to ∞) [(-1)ⁿ tⁿ],0 < t < 1
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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
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
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
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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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