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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 10.8.15
Chapter 10, Problem 10.8.15

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) (−7)ᵏ / k!

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Identify the given series: \( \sum_{k=1}^{\infty} \frac{(-7)^k}{k!} \). This is an infinite series with terms involving factorials in the denominator.
Recall that factorials grow very rapidly, which often suggests the series might converge. To confirm, consider applying the Ratio Test, which is useful for series with factorials and exponentials.
Set up the Ratio Test by examining the limit \( L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| \), where \( a_k = \frac{(-7)^k}{k!} \). Substitute to get \( L = \lim_{k \to \infty} \left| \frac{(-7)^{k+1} / (k+1)!}{(-7)^k / k!} \right| \).
Simplify the expression inside the limit: \( L = \lim_{k \to \infty} \left| \frac{-7}{k+1} \right| = \lim_{k \to \infty} \frac{7}{k+1} \). Since \( \frac{7}{k+1} \to 0 \) as \( k \to \infty \), the limit \( L = 0 \).
Interpret the Ratio Test result: since \( L < 1 \), the series \( \sum_{k=1}^{\infty} \frac{(-7)^k}{k!} \) converges absolutely. Therefore, the series 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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Ratio Test

The ratio test evaluates the limit of the absolute value of the ratio of consecutive terms. If this limit is less than one, the series converges absolutely; if greater than one, it diverges. This test is particularly useful for series involving factorials or exponential terms.
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Factorials and Their Growth

Factorials (n!) grow very rapidly compared to exponential or polynomial terms. Recognizing the dominance of factorial growth helps in applying convergence tests, as terms with factorials in the denominator often lead to convergence due to rapid term size reduction.
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Related Practice
Textbook Question

"21–26. Recurrence relations Write the first four terms of the sequence {aₙ} defined by the following recurrence relations.

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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 ∞) ((k / (k + 1)) × 2k²)

Textbook Question

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


∑ (from k = 1 to ∞) (−k)³ / (3k³ + 2)

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⁺² / 5ᵏ

Textbook Question

45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.

∑ (k = 1 to ∞) (−1)ᵏ · tan⁻¹(k) / k³

Textbook Question

8–32. {Use of Tech} Estimating errors in partial sums For each of the following convergent alternating series, evaluate the nth partial sum for the given value of n. Then use Theorem 10.18 to find an upper bound for the error |S − Sₙ| in using the nth partial sum Sₙ to estimate the value of the series S.


∑ (k = 1 to ∞) (−1)ᵏ / k⁴; n = 4

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