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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 10.5.59
Chapter 10, Problem 10.5.59

40–62. Choose your test Use the test of your choice to determine whether the following series converge.
∑ (k = 1 to ∞) tan(1 / k)

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Identify the series given: \( \sum_{k=1}^{\infty} \tan\left(\frac{1}{k}\right) \). We want to determine if this series converges or diverges.
Recall that for large \( k \), \( \frac{1}{k} \) approaches 0, so consider the behavior of \( \tan(x) \) as \( x \to 0 \). Use the fact that \( \tan(x) \approx x \) when \( x \) is close to 0.
Approximate the terms of the series for large \( k \): \( \tan\left(\frac{1}{k}\right) \approx \frac{1}{k} \). This suggests the series behaves similarly to the harmonic series \( \sum \frac{1}{k} \).
Since the harmonic series \( \sum \frac{1}{k} \) diverges, use the Limit Comparison Test with \( a_k = \tan\left(\frac{1}{k}\right) \) and \( b_k = \frac{1}{k} \). Compute \( \lim_{k \to \infty} \frac{a_k}{b_k} \).
If the limit is a finite nonzero number, then both series either converge or diverge together. Since \( \sum \frac{1}{k} \) diverges, conclude the behavior of the original series accordingly.

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Key Concepts

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

Convergence of Infinite Series

An infinite series converges if the sequence of its partial sums approaches a finite limit. Understanding convergence is essential to determine whether the sum of infinitely many terms results in a finite value or diverges to infinity or oscillates.
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Convergence of an Infinite Series

Comparison and Limit Comparison Tests

These tests compare the given series to a known benchmark series to determine convergence. The limit comparison test involves taking the limit of the ratio of terms from two series, helping to conclude convergence or divergence based on the behavior of the simpler series.
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Limit Comparison Test

Behavior of tan(1/k) for Large k

As k approaches infinity, 1/k approaches zero, and tan(1/k) behaves similarly to 1/k because tan(x) ~ x near zero. Recognizing this helps approximate the series terms and apply appropriate convergence tests by comparing to the harmonic series.
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Newton's Law of Cooling Example 5
Related Practice
Textbook Question

Periodic doses

Suppose you take 200 mg of an antibiotic every 6 hr. The half-life of the drug (the time it takes for half of the drug to be eliminated from your blood) is 6 hr. Use infinite series to find the long-term (steady-state) amount of antibiotic in your blood. You may assume the steady-state amount is finite.

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Textbook Question

40–62. Choose your test Use the test of your choice to determine whether the following series converge.

∑ (k = 2 to ∞) (5lnk) / k

Textbook Question

9–15. Geometric sums Evaluate each geometric sum.


{Use of Tech}∑ k = 0 to 9(−3/4)ᵏ

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Textbook Question

61–66. Sequences of partial sums For the following infinite series, find the first four terms of the sequence of partial sums. Then make a conjecture about the value of the infinite series or state that the series diverges.


4 + 0.9 + 0.09 + 0.009 + ⋯  

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Textbook Question

45–48. {Use of Tech} Explicit formulas for sequences Consider the formulas for the following sequences {aₙ}ₙ₌₁∞

 Make a table with at least ten terms and determine a plausible limit of the sequence or state that the sequence diverges.

aₙ = ⁿ² + n ;n = 1, 2, 3, …

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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.

∑ (from k = 1 to ∞) 2ᵏ k! / kᵏ