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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 10.7.11
Capitolo 10, Problema 10.7.11

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 ∞) ((-1)ᵏ⁺¹) × ((10k³ + k) / (9k³ + k + 1))ᵏ

Guida verificata passo dopo passo
1
Identify the general term of the series: \(a_k = (-1)^{k+1} \left( \frac{10k^3 + k}{9k^3 + k + 1} \right)^k\).
Since the series has terms raised to the power \(k\), the Root Test is a natural choice. Recall the Root Test uses the limit \(L = \lim_{k \to \infty} \sqrt[k]{|a_k|}\).
Calculate \(\sqrt[k]{|a_k|}\). Because of the absolute value, the \((-1)^{k+1}\) disappears, so \(\sqrt[k]{|a_k|} = \sqrt[k]{\left( \frac{10k^3 + k}{9k^3 + k + 1} \right)^k} = \frac{10k^3 + k}{9k^3 + k + 1}\).
Evaluate the limit \(L = \lim_{k \to \infty} \frac{10k^3 + k}{9k^3 + k + 1}\). To do this, divide numerator and denominator by \(k^3\) to simplify the expression.
Interpret the result of the limit \(L\): if \(L < 1\), the series converges absolutely; if \(L > 1\), the series diverges; if \(L = 1\), the Root Test is inconclusive.

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Ratio Test

The Ratio Test determines the convergence of a 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 the nth root of the absolute value of the nth term of a series. If the limit of this root is less than 1, the series converges absolutely; if greater than 1, it diverges; if equal to 1, the test is inconclusive. It is especially useful for terms raised to the nth power.
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Percorso guidato
07:15
Root Test

Absolute Convergence

A series converges absolutely if the series of absolute values of its terms converges. Absolute convergence implies convergence regardless of term signs, which is important when applying tests like the Ratio or Root Test to series with alternating signs.
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Percorso guidato
07:51
Choosing a Convergence Test
Pratica correlata
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40–62. Choose your test Use the test of your choice to determine whether the following series converge.

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13–52. Limits of sequences

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


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

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


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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.

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