교과서 질문
Use the Integral Test to show that the series
∑ (from n=0 to ∞) e^(−n²)
converges.
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 10.2.98
Find the value of b for which
1 + eᵇ + e²ᵇ + e³ᵇ + ⋯ = 9.
검증된 단계별 안내1
Recognize that the given series is an infinite geometric series of the form \(1 + e^{b} + e^{2b} + e^{3b} + \cdots\).
Recall the formula for the sum of an infinite geometric series \(S = \frac{a}{1 - r}\), where \(a\) is the first term and \(r\) is the common ratio, with the condition \(|r| < 1\) for convergence.
Identify the first term \(a = 1\) and the common ratio \(r = e^{b}\) in the series.
Set up the equation for the sum: \(\frac{1}{1 - e^{b}} = 9\).
Solve this equation for \(b\) by isolating \(e^{b}\) and then taking the natural logarithm to find \(b\).
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.
Geometric Series
A geometric series is a sum of terms where each term is found by multiplying the previous term by a constant ratio. For example, the series 1 + r + r² + r³ + ⋯ converges if |r| < 1, and its sum can be calculated using a specific formula.
추천 영상:
가이드 코스
06:00
Geometric Series
Sum of an Infinite Geometric Series
If the common ratio r of a geometric series satisfies |r| < 1, the infinite sum converges to S = 1 / (1 - r). This formula allows us to find the sum of infinitely many terms without adding them individually.
추천 영상:
가이드 코스
06:00Geometric Series
Solving Exponential Equations
Solving exponential equations involves isolating the variable in the exponent. This often requires applying logarithms or algebraic manipulation to rewrite the equation in a solvable form.
추천 영상:
5:47Solving Exponential Equations Using Logs
관련 실천
교과서 질문
Determining Convergence or Divergence
In Exercises 17–46, use any method to determine whether the series converges or diverges. Give reasons for your answer.
∑(from n=1 to ∞) [(2 + (−1)ⁿ) / 1.25ⁿ]
교과서 질문
In Exercises 57–82, use any method to determine whether the series converges or diverges. Give reasons for your answer.
∑ (from n = 0 to ∞) [(2n + 3)(2ⁿ + 3) / (3ⁿ + 2)]
교과서 질문
Using the Root Test
In Exercises 9–16, use the Root Test to determine if each series converges absolutely or diverges.
∑(from n=1 to ∞) [4ⁿ / (3n)ⁿ]
56
views
교과서 질문
Uniqueness of limits Prove that limits of sequences are unique. That is, show that if L₁ and L₂ are numbers such that aₙ → L₁ and aₙ → L₂, then L₁ = L₂.
32
views
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 = (n + 3) / (n² + 5n + 6)
29
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