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 ∞) [ (ln n) xⁿ ]
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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 10.2.30
Repeating Decimals
Express each of the numbers in Exercises 23–30 as the ratio of two integers.
3.1̅4̅2̅8̅5̅7 = 3.142857142857 ...
검증된 단계별 안내1
Identify the repeating decimal part of the number. Here, the decimal 3.1̅4̅2̅8̅5̅7̅ means the digits 142857 repeat indefinitely after the decimal point.
Let \( x = 3.142857142857... \). Since the repeating block has 6 digits, multiply both sides by \( 10^6 = 1,000,000 \) to shift the decimal point 6 places to the right: \( 10^6 x = 3142857.142857... \).
Subtract the original number \( x \) from this new equation to eliminate the repeating decimal part: \( 10^6 x - x = 3142857.142857... - 3.142857... \).
Simplify the subtraction to get \( (10^6 - 1) x = 3142854 \), which is \( 999999 x = 3142854 \).
Solve for \( x \) by dividing both sides by 999999: \( x = \frac{3142854}{999999} \). This fraction represents the repeating decimal as a ratio of two integers.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
2m주요 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, 3.142857142857... has the repeating block '142857'. Recognizing the repeating pattern is essential to convert the decimal into a fraction.
추천 영상:
가이드 코스
08:23
Repeated Integration by Parts
Conversion of Repeating Decimals to Fractions
To express a repeating decimal as a ratio of two integers, set the decimal equal to a variable, multiply to shift the decimal point past the repeating block, and subtract to eliminate the repeating part. Solving the resulting equation yields the fraction form.
추천 영상:
가이드 코스
11:26Partial Fraction Decomposition: Repeated Linear Factors
Properties of Rational Numbers
Rational numbers are numbers that can be expressed as the ratio of two integers. Every repeating decimal corresponds to a rational number, so understanding this property helps in confirming that the decimal can be converted into a fraction.
추천 영상:
가이드 코스
06:21Properties of Functions
관련 실천
교과서 질문
Convergence and Divergence
Which of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.
aₙ = (xⁿ / (2n + 1))^(1/n),x > 0
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교과서 질문
In Exercises 43–50, use Theorem 20 to find the series’ interval of convergence and, within this interval, the sum of the series as a function of x.
∑ (from n = 0 to ∞) [ (x² − 1) / 2 ]ⁿ
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교과서 질문
Finding a Sequence’s Formula
In Exercises 13–30, find a formula for the nth term of the sequence.
1, -4, 9, -16, 25, …Squares of the positive integers, with alternating signs
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교과서 질문
In Exercises 37–42, find the series’ radius of convergence.
∑ (from n = 1 to ∞) [ ( n / (n + 1) )ⁿ^ ² ] xⁿ (Hint: Apply the Root Test.)
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교과서 질문
Which series in Exercises 53–76 converge, and which diverge? Give reasons for your answers. If a series converges, find its sum.
∑ (from n = 1 to ∞) (1 − 1/n)ⁿ
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