Textbook Question
21–42. Geometric series Evaluate each geometric series or state that it diverges.
23.∑ (k = 0 to ∞) (–9/10)ᵏ
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 10.2.65
55–70. More sequences
Find the limit of the following sequences or determine that the sequence diverges.
{cosn / n}
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Identify the given sequence as \( a_n = \frac{\cos n}{n} \), where \( n \) is a positive integer.
Recall that \( \cos n \) oscillates between -1 and 1 for all integer values of \( n \), so the numerator is bounded.
Note that the denominator \( n \) increases without bound as \( n \to \infty \).
Use the Squeeze Theorem by bounding the sequence: since \( -1 \leq \cos n \leq 1 \), we have \( -\frac{1}{n} \leq \frac{\cos n}{n} \leq \frac{1}{n} \).
Since both \( -\frac{1}{n} \) and \( \frac{1}{n} \) approach 0 as \( n \to \infty \), conclude that \( \lim_{n \to \infty} \frac{\cos n}{n} = 0 \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limit of a Sequence
The limit of a sequence is the value that the terms of the sequence approach as the index n goes to infinity. If the terms get arbitrarily close to a specific number, the sequence converges to that limit; otherwise, it diverges.
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Introduction to Sequences
Behavior of the Cosine Function
The cosine function oscillates between -1 and 1 for all real numbers n. Since cos(n) does not approach a single value as n increases, it is important to consider how this oscillation affects the sequence when combined with other terms.
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5:53Graph of Sine and Cosine Function
Squeeze Theorem for Sequences
The Squeeze Theorem states that if a sequence is bounded above and below by two sequences that both converge to the same limit, then the original sequence also converges to that limit. This is useful when dealing with sequences involving bounded oscillating functions divided by terms that grow without bound.
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Related Practice
Textbook Question
33–38. {Use of Tech} Remainders in alternating series Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than 10⁻⁴ in magnitude. Although you do not need it, the exact value of the series is given in each case.
ln 2 = ∑ (k = 1 to ∞) (−1)ᵏ⁺¹ / k
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Textbook Question
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞)(1 + 1 / (2k))ᵏ
Textbook Question
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞)sin(1 / k⁹)
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).
51.0.456̅ = 0.456456456…
Textbook Question
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞)tan⁻¹(1 / √k)