Textbook Question
Growth rates of sequences
Use Theorem 10.6 to find the limit of the following sequences or state that they diverge.
aₙ = (6ⁿ + 3ⁿ) / (6ⁿ + n¹⁰⁰)
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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
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 10, Problem 10.3.35
21–42. Geometric series Evaluate each geometric series or state that it diverges.
35.∑ (k = 0 to ∞) 3(–π)^(–k)
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Identify the first term \( a \) of the geometric series by substituting \( k = 0 \) into the general term \( 3(-\pi)^{-k} \). This gives \( a = 3(-\pi)^0 = 3 \).
Determine the common ratio \( r \) by finding the factor that each term is multiplied by to get the next term. This is \( r = \frac{3(-\pi)^{-1}}{3(-\pi)^0} = (-\pi)^{-1} = \frac{1}{-\pi} = -\frac{1}{\pi} \).
Check the convergence of the series by evaluating the absolute value of the common ratio \( |r| = \left| -\frac{1}{\pi} \right| = \frac{1}{\pi} \). Since \( \pi > 1 \), \( |r| < 1 \), so the series converges.
Use the formula for the sum of an infinite geometric series \( S = \frac{a}{1 - r} \) to express the sum, where \( a = 3 \) and \( r = -\frac{1}{\pi} \).
Write the sum explicitly as \( S = \frac{3}{1 - \left(-\frac{1}{\pi}\right)} = \frac{3}{1 + \frac{1}{\pi}} \). Simplify this expression to get the sum in a more compact form.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Geometric Series Definition
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 ∑ ar^k, where a is the first term and k is the index. Understanding this structure is essential to identify and evaluate the series.
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Convergence Criteria for Infinite Geometric Series
An infinite geometric series converges if and only if the absolute value of the common ratio |r| is less than 1. If |r| ≥ 1, the series diverges. This criterion determines whether the sum approaches a finite value or not.
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Sum Formula for Convergent Geometric Series
When 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 quick way to find the total sum of infinitely many terms.
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Related Practice
Textbook Question
40–62. Choose your test Use the test of your choice to determine whether the following series converge.
∑ (k = 1 to ∞) (1 + 2 / k)ᵏ
Textbook Question
72–86. Evaluating series Evaluate each series or state that it diverges.
∑ (k = 0 to ∞) (1/4)ᵏ × 5^(3 – k)
Textbook Question
45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (k = 2 to ∞) (−1)ᵏ · k · (k² + 1) / (k³ − 1)
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Textbook Question
48–63. Choose your test Determine whether the following series converge or diverge using the properties and tests introduced in Sections 10.3 and 10.4.
∑ (k = 2 to ∞) 1 / eᵏ
Textbook Question
Find two different explicit formulas for the sequence {1, -2, 3, -4, -5 .....}
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