Textbook Question
In Exercises 67–72, use the results of Exercises 63 and 64 to determine if each series converges or diverges.
∑(from n=2 to ∞) [(ln n)¹⁰⁰⁰ / n¹.⁰⁰¹]
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Ch. 10 - Infinite Sequences and Series
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- 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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 10, Problem 10.2.98
Find the value of b for which
1 + eᵇ + e²ᵇ + e³ᵇ + ⋯ = 9.
Verified step by step guidance1
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 \(b\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
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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.
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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.
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Related Practice
Textbook Question
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 = 0 to ∞) [ n xⁿ / (4ⁿ (n² + 1)) ]
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Textbook Question
Applying the Integral Test
Use the Integral Test to determine if the series in Exercises 1–12 converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied.
∑ (from n = 2 to ∞) ln(n²) / n
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Textbook Question
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.
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Textbook Question
Use power series operations to find the Taylor series at x = 0 for the functions in Exercises 13–30.
sin x – x + (x³ / 3!)
Textbook Question
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)
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