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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 10.8.53
Chapter 10, Problem 10.8.53

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞)sin(1 / k⁹)

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Identify the series given: \( \sum_{k=1}^{\infty} \sin\left(\frac{1}{k^9}\right) \). We want to determine if this series converges.
Recall that for very small angles \( x \), \( \sin x \approx x \). Since \( \frac{1}{k^9} \) becomes very small as \( k \to \infty \), we can approximate \( \sin\left(\frac{1}{k^9}\right) \approx \frac{1}{k^9} \) for large \( k \).
Compare the given series to the p-series \( \sum_{k=1}^{\infty} \frac{1}{k^p} \) where \( p = 9 \). We know that a p-series converges if \( p > 1 \). Since 9 is much greater than 1, the p-series \( \sum \frac{1}{k^9} \) converges.
Use the Limit Comparison Test to justify convergence: compute \( \lim_{k \to \infty} \frac{\sin\left(\frac{1}{k^9}\right)}{\frac{1}{k^9}} \). If this limit is a finite nonzero number, then both series behave similarly in terms of convergence.
Since the limit comparison test shows the given series behaves like a convergent p-series, conclude that \( \sum_{k=1}^{\infty} \sin\left(\frac{1}{k^9}\right) \) converges.

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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 Test for Series

The comparison test involves comparing the given series to a known benchmark series. If the terms of the given series are smaller than those of a convergent series, it also converges; if larger than a divergent series, it diverges. This test helps analyze series with complicated terms.
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Direct Comparison Test

Behavior of sin(x) for Small x

For values of x close to zero, sin(x) is approximately equal to x. This approximation allows simplification of terms like sin(1/k⁹) to 1/k⁹ for large k, facilitating the use of p-series tests to determine convergence.
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Integrals of Natural Exponential Functions (e^x)
Related Practice
Textbook Question

21–42. Geometric series Evaluate each geometric series or state that it diverges.  


23.∑ (k = 0 to ∞) (–9/10)ᵏ

Textbook Question

21–42. Geometric series Evaluate each geometric series or state that it diverges.  


41.∑ (k = 1 to ∞) 4 / 12ᵏ

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

55–70. More sequences

Find the limit of the following sequences or determine that the sequence diverges.


{cosn / n}

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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…