교과서 질문
13–20. Explicit formulas Write the first four terms of the sequence { aₙ }∞ₙ₌₁.
aₙ = (−1)ⁿ / 2ⁿ
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Ch. 10 - Sequences and Infinite Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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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 10.8.31
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from j = 1 to ∞) 5 / (j² + 4)
검증된 단계별 안내1
Identify the given series: \( \sum_{j=1}^{\infty} \frac{5}{j^{2} + 4} \). We want to determine if this series converges or diverges.
Recognize that the terms \( \frac{5}{j^{2} + 4} \) are positive and decrease as \( j \) increases, since the denominator grows quadratically.
Compare the given series to a known benchmark series. Notice that \( \frac{5}{j^{2} + 4} < \frac{5}{j^{2}} \) for all \( j \geq 1 \). The series \( \sum_{j=1}^{\infty} \frac{5}{j^{2}} \) is a constant multiple of the p-series \( \sum \frac{1}{j^{2}} \) with \( p = 2 > 1 \), which is known to converge.
Apply the Comparison Test: since \( \sum \frac{5}{j^{2}} \) converges and \( \frac{5}{j^{2} + 4} \leq \frac{5}{j^{2}} \), the original series \( \sum \frac{5}{j^{2} + 4} \) also converges.
Conclude that the series converges by the Comparison Test, justifying the answer based on the behavior of the terms and the known convergence of the p-series.
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 sequence of its partial sums approaches a finite limit. Understanding convergence is essential to determine whether the sum of infinitely many terms results in a finite value or diverges to infinity or oscillates.
추천 영상:
가이드 코스
06:52
Convergence of an Infinite Series
Comparison Test
The Comparison Test involves comparing a given series to a second series whose convergence behavior is known. If the terms of the given series are smaller than those of a convergent series, it also converges; if larger than a divergent series, it diverges.
추천 영상:
가이드 코스
09:25Direct Comparison Test
p-Series Test
A p-series is of the form ∑ 1/n^p, which converges if p > 1 and diverges otherwise. Recognizing that the given series resembles a p-series helps in applying this test to determine convergence by comparing the denominator's growth rate.
추천 영상:
가이드 코스
04:30P-Series and Harmonic Series
관련 실천
교과서 질문
40–62. Choose your test Use the test of your choice to determine whether the following series converge.
∑ (k = 3 to ∞) 1 / lnk
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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/k)
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교과서 질문
54–69. Telescoping series
For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sₙ}. Then evaluate limₙ→∞ Sₙ to obtain the value of the series or state that the series diverges.
61. ∑ (k = 1 to ∞) ln((k + 1) / k)
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교과서 질문
Why does the value of a converging alternating series with terms that are nonincreasing in magnitude lie between any two consecutive terms of its sequence of partial sums?
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교과서 질문
17–22. Integral Test Use the Integral Test to determine whether the following series converge after showing that the conditions of the Integral Test are satisfied.
∑ (k = 1 to ∞) 1 / (∛(5k + 3))
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