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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblema 10.7.23
Capitolo 10, Problema 10.7.23

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)

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Identify the general term of the series: \(a_k = \left( \frac{k}{k+1} \right) \times 2^{k^2}\).
Recall the Ratio Test: For a series \(\sum a_k\), compute the limit \(L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right|\). If \(L < 1\), the series converges absolutely; if \(L > 1\), it diverges; if \(L = 1\), the test is inconclusive.
Compute the ratio \(\frac{a_{k+1}}{a_k} = \frac{\left( \frac{k+1}{k+2} \right) 2^{(k+1)^2}}{\left( \frac{k}{k+1} \right) 2^{k^2}} = \left( \frac{k+1}{k+2} \times \frac{k+1}{k} \right) \times 2^{(k+1)^2 - k^2}\).
Simplify the exponent difference: \((k+1)^2 - k^2 = 2k + 1\), so the ratio becomes \(\left( \frac{(k+1)^2}{k(k+2)} \right) \times 2^{2k+1}\).
Evaluate the limit as \(k \to \infty\) of the ratio. Since \(2^{2k+1}\) grows exponentially, the limit will be infinite, which implies by the Ratio Test that the series diverges.

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Ratio Test

The Ratio Test determines the convergence of an infinite series by examining the limit of the absolute value of the ratio of consecutive terms. If this limit is less than 1, the series converges absolutely; if greater than 1, it diverges; if equal to 1, the test is inconclusive.
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Percorso guidato
05:35
Ratio Test

Root Test

The Root Test analyzes convergence by taking the nth root of the absolute value of the nth term of a series. If the limit of this root as n approaches infinity is less than 1, the series converges absolutely; if greater than 1, it diverges; if equal to 1, the test is inconclusive.
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Percorso guidato
07:15
Root Test

Absolute Convergence

A series converges absolutely if the series of the absolute values of its terms converges. Absolute convergence guarantees convergence regardless of term signs, making tests like the Ratio and Root Tests effective tools for determining the behavior of series with positive or negative terms.
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07:51
Choosing a Convergence Test
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13–52. Limits of sequences

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


{tan⁻¹(10n⁄(10n + 4))}  

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Given the series ∑∞ₖ₌₁ k, evaluate the first four terms of its sequence of partial sums Sₙ = ∑ⁿₖ₌₁ k. 

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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 = 4 to ∞) (1 + cos²(k)) / (k − 3)

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


∑ (from k = 1 to ∞) (−k)³ / (3k³ + 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 ∞) 3ᵏ⁺² / 5ᵏ

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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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