교과서 질문
45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (k = 1 to ∞) (−1)ᵏ / k^(2/3)
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Ch. 10 - Sequences and Infinite Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- 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 10.3.49
46–53. Decimal expansions
Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers).
49.0.037̅ = 0.037037…
검증된 단계별 안내1
Identify the repeating decimal part. Here, the repeating block is "037", which repeats indefinitely after the decimal point.
Express the decimal as a sum of its non-repeating part plus the repeating part written as an infinite geometric series. Since the repeating block "037" starts immediately after the decimal, write it as: \(0.037 + 0.000037 + 0.000000037 + \cdots\).
Rewrite the repeating part as a geometric series with the first term \(a = 0.037\) and common ratio \(r = 10^{-3}\) (because the repeating block has 3 digits, so each term is shifted by 3 decimal places). The series is \(a + ar + ar^2 + \cdots\).
Use the formula for the sum of an infinite geometric series \(S = \frac{a}{1 - r}\) to find the sum of the repeating part.
Combine the sum of the geometric series with any non-repeating part (if any) and simplify the expression to write the decimal as a fraction (ratio of two integers).
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Repeating Decimals
A repeating decimal is a decimal number in which a sequence of digits repeats infinitely. For example, 0.037037… has the block '037' repeating endlessly. Understanding this pattern is essential to express the decimal as a series or fraction.
추천 영상:
08:23
Repeated Integration by Parts
Geometric Series
A geometric series is a sum of terms where each term is found by multiplying the previous term by a constant ratio. Repeating decimals can be represented as infinite geometric series by identifying the repeating block as the first term and the power of 10 as the common ratio.
추천 영상:
06:00Geometric Series
Conversion of Geometric Series to Fraction
An infinite geometric series with first term a and common ratio r (|r|<1) sums to a/(1-r). Using this formula, the repeating decimal's geometric series can be converted into a fraction, expressing the decimal as a ratio of two integers.
추천 영상:
06:00Geometric Series
관련 실천
교과서 질문
21–42. Geometric series Evaluate each geometric series or state that it diverges.
23.∑ (k = 0 to ∞) (–9/10)ᵏ
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교과서 질문
9–30. The Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge.
∑ (from k = 1 to ∞) ((-1)ᵏ) / (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.
57. ∑ (k = 1 to ∞) 1 / ((k + 6)(k + 7))
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교과서 질문
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) (10ᵏ + 1) / k¹⁰
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교과서 질문
1–10. Choosing convergence tests Identify a convergence test for each series. If necessary, explain how to simplify or rewrite the series before applying the convergence test. You do not need to carry out the convergence test.
∑ (from k = 1 to ∞) ((−1)ᵏ⁺¹) / (√2ᵏ + lnk)
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