교과서 질문
Make up an infinite series of nonzero terms whose sum is
b. −3
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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 10.2.19
In Exercises 15–22, determine if the geometric series converges or diverges. If a series converges, find its sum.
1 − (2/e) + (2/e)² − (2/e)³ + (2/e)⁴ − …
검증된 단계별 안내1
Identify the first term \( a \) and the common ratio \( r \) of the geometric series. The series is \( 1 - \frac{2}{e} + \left(\frac{2}{e}\right)^2 - \left(\frac{2}{e}\right)^3 + \left(\frac{2}{e}\right)^4 - \cdots \), so \( a = 1 \) and \( r = -\frac{2}{e} \).
Recall that a geometric series \( \sum_{n=0}^\infty ar^n \) converges if and only if the absolute value of the common ratio is less than 1, i.e., \( |r| < 1 \).
Calculate \( |r| = \left| -\frac{2}{e} \right| = \frac{2}{e} \). Since \( e \approx 2.718 \), compare \( \frac{2}{e} \) to 1 to determine convergence.
If the series converges, use the formula for the sum of an infinite geometric series: \[ S = \frac{a}{1 - r} \]. Substitute the values of \( a \) and \( r \) into this formula.
Write the sum expression explicitly as \( S = \frac{1}{1 - \left(-\frac{2}{e}\right)} = \frac{1}{1 + \frac{2}{e}} \), which can be simplified further if needed.
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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Geometric Series
A geometric series is a sum of terms where each term is found by multiplying the previous term by a constant ratio, r. It has the form a + ar + ar² + ar³ + ..., where a is the first term. Understanding this structure is essential to analyze the given series.
추천 영상:
가이드 코스
06:00
Geometric Series
Convergence Criteria for Geometric Series
A geometric series converges if the absolute value of the common ratio |r| is less than 1. If |r| ≥ 1, the series diverges. This criterion helps determine whether the infinite sum approaches a finite value.
추천 영상:
가이드 코스
06:00Geometric Series
Sum of a Convergent Geometric Series
If a geometric series converges, its sum can be calculated using the formula S = a / (1 - r), where a is the first term and r is the common ratio. This formula provides a direct way to find the total sum of the infinite series.
추천 영상:
가이드 코스
06:00Geometric Series
관련 실천
교과서 질문
Use series to evaluate the limits in Exercises 29–40.
29. lim (x → 0) (e^x - (1 + x)) / x²
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교과서 질문
Determining Convergence or Divergence
Which of the series in Exercises 13–46 converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)
∑ (from n=1 to ∞) eⁿ / (1 + e²ⁿ)
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교과서 질문
Finding Terms of a Sequence
Each of Exercises 1–6 gives a formula for the nth term aₙ of a sequence {aₙ}. Find the values of a₁, a₂, a₃, and a₄.
aₙ = 1 / n!
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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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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
28
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