Textbook Question
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞)(cos(1 / k) – cos(1 / (k + 1)))
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 10.3.27
21–42. Geometric series Evaluate each geometric series or state that it diverges.
27.1 + 1.01 + 1.01² + 1.01³ + ⋯
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Identify the first term \( a \) and the common ratio \( r \) of the geometric series. Here, the first term is \( a = 1 \) and the common ratio is \( r = 1.01 \).
Recall that a geometric series \( a + ar + ar^2 + ar^3 + \cdots \) converges if and only if the absolute value of the common ratio \( |r| < 1 \).
Check the value of \( |r| \). Since \( r = 1.01 \), which is greater than 1, the series does not satisfy the convergence condition.
Conclude that because \( |r| > 1 \), the geometric series diverges and does not have a finite sum.
Therefore, the series \( 1 + 1.01 + 1.01^2 + 1.01^3 + \cdots \) diverges and cannot be evaluated to a finite value.
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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. It has the form a + ar + ar² + ar³ + ⋯, where a is the first term and r is the common ratio.
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Geometric Series
Convergence and Divergence of Geometric Series
A geometric series converges if the absolute value of the common ratio |r| is less than 1, meaning the terms get smaller and approach zero. If |r| ≥ 1, the series diverges and does not have a finite sum.
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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 the finite sum of infinitely many terms.
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Related Practice
Textbook Question
46–53. Decimal expansions
Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers).
49.0.037̅ = 0.037037…
Textbook Question
45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (k = 1 to ∞) (−1)ᵏ⁺¹ / (2√k − 1)
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 = 1 to ∞) k / √(k² + 1)
Textbook Question
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{(3n³ − 1)⁄(2n³ + 1)}
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Textbook Question
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from j = 1 to ∞)cot(–1 / j) / 2ʲ