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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblem 10.2.19
Chapter 10, Problem 10.2.19

13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.


{1 + cos(1⁄n)}

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1
Identify the given sequence: \(a_n = 1 + \cos\left(\frac{1}{n}\right)\), where \(n\) is a positive integer and \(n \to \infty\).
Recall that as \(n\) approaches infinity, the term \(\frac{1}{n}\) approaches 0, so we need to analyze the behavior of \(\cos\left(\frac{1}{n}\right)\) as its argument approaches 0.
Use the fact that \(\cos(x)\) is continuous and \(\cos(0) = 1\), so \(\lim_{x \to 0} \cos(x) = 1\). Therefore, \(\lim_{n \to \infty} \cos\left(\frac{1}{n}\right) = 1\).
Apply the limit to the entire sequence: \(\lim_{n \to \infty} a_n = \lim_{n \to \infty} \left(1 + \cos\left(\frac{1}{n}\right)\right) = 1 + \lim_{n \to \infty} \cos\left(\frac{1}{n}\right)\).
Combine the results to conclude that the limit of the sequence is \(1 + 1 = 2\).

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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 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 Near Zero

The cosine function is continuous and approaches 1 as its argument approaches 0. Understanding that cos(1/n) approaches cos(0) = 1 as n → ∞ is key to evaluating the limit of sequences involving cosine of reciprocal terms.
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Graph of Sine and Cosine Function

Substitution and Limit Laws

Limit laws allow the evaluation of limits by substituting the limit of the inner function into the outer function when the outer function is continuous. Here, since cosine is continuous, we can find the limit by substituting the limit of 1/n into cos(1/n).
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Related Practice
Textbook Question

32–49. Choose your test Use the test of your choice to determine whether the following series converge absolutely, converge conditionally, or diverge.

1 / 1! + 4 / 2! + 9 / 3! + 16 / 4! + ⋯

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 ∞) 3ᵏ / (k² + 1)

Textbook Question

23–38. Divergence, Integral, and p-series Tests Use the Divergence Test, the Integral Test, or the p-series test to determine whether the following series converge.

∑ (k = 3 to ∞) 1 / (k − 2)⁴

Textbook Question

9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.

∑ (k = 1 to ∞) 1 / (k² + 4)

Textbook Question

54–69. Telescoping series

For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sₙ}. Then evaluate limₙ→∞ Sₙ to obtain the value of the series or state that the series diverges.


67. ∑ (k = 1 to ∞) 3 / (k² + 5k + 4)

Textbook Question

For what values of p does the series ∑ (k = 10 to ∞) 1 / kᵖ converge (initial index is 10)? For what values of p does it diverge?