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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblema 10.8.51
Capitolo 10, Problema 10.8.51

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

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Rewrite the general term of the series to simplify it. The term is given by \( \frac{7^k + 11^k}{11^k} \). Split this into two separate fractions: \( \frac{7^k}{11^k} + \frac{11^k}{11^k} \).
Simplify each fraction: \( \frac{7^k}{11^k} = \left(\frac{7}{11}\right)^k \) and \( \frac{11^k}{11^k} = 1 \). So the general term becomes \( \left(\frac{7}{11}\right)^k + 1 \).
Analyze the behavior of the terms as \( k \to \infty \). Since \( \left(\frac{7}{11}\right)^k \to 0 \), the term approaches \( 1 \).
Recall the necessary condition for series convergence: if \( \lim_{k \to \infty} a_k \neq 0 \), then the series \( \sum a_k \) diverges. Here, the limit of the terms is 1, not 0.
Conclude that the series \( \sum_{k=1}^\infty \frac{7^k + 11^k}{11^k} \) diverges because its terms do not approach zero.

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Infinite Series and Convergence

An infinite series is the sum of infinitely many terms. Determining convergence means checking if the sum approaches a finite limit as the number of terms grows. Understanding the behavior of the series' terms is essential to decide if the series converges or diverges.
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Convergence of an Infinite Series

Geometric Series

A geometric series has terms that form a constant ratio with the previous term. It converges if the absolute value of the common ratio is less than one, and its sum can be calculated using a specific formula. Recognizing geometric series helps simplify and analyze series involving exponential terms.
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Geometric Series

Convergence Tests (Comparison and Limit Tests)

Convergence tests like the Comparison Test and the Limit Test help determine if a series converges by comparing it to known series or analyzing the limit of term ratios. These tests are crucial when series are not straightforward geometric series but can be related to them.
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Limit Comparison Test
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Domanda del libro di testo

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

aₙ₊₁ = 2aₙ; a₁ = 2

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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 ∞) 1 / ( (3k + 1)(3k + 4) )

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


∑ (k = 1 to ∞) 20 / (∛k + √k)

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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! + ⋯

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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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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?

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