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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 10.8.65
Chapter 10, Problem 10.8.65

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)))

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Recognize that the given series is \( \sum_{k=1}^{\infty} \left( \cos\left(\frac{1}{k}\right) - \cos\left(\frac{1}{k+1}\right) \right) \), which is a telescoping series because each term involves the difference of cosine values at consecutive indices.
Write out the first few partial sums explicitly to observe the telescoping pattern: \( S_n = \sum_{k=1}^n \left( \cos\left(\frac{1}{k}\right) - \cos\left(\frac{1}{k+1}\right) \right) = \left( \cos(1) - \cos\left(\frac{1}{2}\right) \right) + \left( \cos\left(\frac{1}{2}\right) - \cos\left(\frac{1}{3}\right) \right) + \cdots + \left( \cos\left(\frac{1}{n}\right) - \cos\left(\frac{1}{n+1}\right) \right) \).
Notice that most terms cancel out in the sum, leaving only the first term of the first cosine and the last term of the last cosine: \( S_n = \cos(1) - \cos\left(\frac{1}{n+1}\right) \).
To determine convergence, analyze the limit of the partial sums as \( n \to \infty \): \( \lim_{n \to \infty} S_n = \lim_{n \to \infty} \left( \cos(1) - \cos\left(\frac{1}{n+1}\right) \right) \).
Since \( \lim_{x \to 0} \cos(x) = 1 \), evaluate the limit to find \( \lim_{n \to \infty} S_n = \cos(1) - 1 \). This limit exists and is finite, so the series converges by the telescoping series test.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Telescoping Series

A telescoping series is one where many terms cancel out when the series is expanded, leaving only a few terms to evaluate the sum. Recognizing this pattern helps simplify the series and determine convergence by examining the behavior of the remaining terms.
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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. For series convergence, understanding the limit of the partial sums or the terms themselves is crucial to determine if the series converges to a finite value.
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Introduction to Sequences

Convergence Tests for Series

Convergence tests, such as the comparison test, ratio test, or root test, help determine whether an infinite series converges or diverges. In this problem, identifying the appropriate test or recognizing the telescoping nature is key to justifying convergence.
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Related Practice
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 ∞) (k² − 1) / (k³ + 4)

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Textbook Question

Suppose the sequence { aₙ} is defined by the explicit formula aₙ = 1/n, for n=1, 2, 3, .....Write out the first five terms of the sequence.

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Textbook Question

43–44. Periodic doses

Suppose you take a dose of m mg of a particular medication once per day. Assume f equals the fraction of the medication that remains in your blood one day later. Just after taking another dose of medication on the second day, the amount of medication in your blood equals the sum of the second dose and the fraction of the first dose remaining in your blood, which is m + mf. Continuing in this fashion, the amount of medication in your blood just after your nth dose is


Aₙ = m + mf + ⋯ + mfⁿ⁻¹.


For the given values of f and m, calculate A₅, A₁₀, A₃₀, and lim (n → ∞) Aₙ. Interpret the meaning of the limit lim (n → ∞) Aₙ.


43.f = 0.25,m = 200 mg

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Textbook Question

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.

∑ (from k = 1 to ∞) 2⁹k / kᵏ

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Textbook Question

9–30. The Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge.

∑ (from k = 1 to ∞) ((-7)ᵏ / k²)

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Textbook Question

72–86. Evaluating series Evaluate each series or state that it diverges.

∑ (k = 1 to ∞) ((1/3) × (5/6)ᵏ + (3/5) × (7/9)ᵏ)

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