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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 10.7.29
Capitolo 10, Problema 10.7.29

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 × k²ᵏ) / (k! × k!)

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1
Identify the general term of the series as \(a_k = \frac{(-1)^{k+1} \cdot k^{2k}}{(k!) \cdot (k!)}\).
Since the series has alternating signs, to check for absolute convergence, consider the absolute value of the terms: \(|a_k| = \frac{k^{2k}}{(k!)^2}\).
Apply the Ratio Test by computing the limit \(L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| = \lim_{k \to \infty} \frac{(k+1)^{2(k+1)}}{((k+1)!)^2} \cdot \frac{(k!)^2}{k^{2k}}\).
Simplify the expression inside the limit by expressing factorials and powers explicitly, for example, \((k+1)! = (k+1) \cdot k!\), and rewrite powers to compare terms.
Evaluate the limit \(L\) and use the Ratio Test criteria: if \(L < 1\), the series converges absolutely; if \(L > 1\), it diverges; if \(L = 1\), the 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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05:35
Ratio Test

Root Test

The Root Test analyzes the nth root of the absolute value of the terms in a series. If the limit of this nth 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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07:15
Root Test

Absolute Convergence

A series converges absolutely if the series of the absolute values of its terms converges. Absolute convergence implies convergence regardless of the sign of terms, which is important when dealing with alternating series like the given one.
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07:51
Choosing a Convergence Test
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Domanda del libro di testo

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


∑ (k = 1 to ∞) (k² − 1) / (k³ + 4)

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Domanda del libro di testo

Suppose the sequence { aₙ} is defined by the explicit formula aₙ = 1/n, for n=1, 2, 3, .....Write out the first five terms of the sequence.

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Domanda del libro di testo

72–86. Evaluating series Evaluate each series or state that it diverges.

∑ (k = 1 to ∞) ((1/3) × (5/6)ᵏ + (3/5) × (7/9)ᵏ)

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Domanda del libro di testo

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.


π / 4 = ∑ (k = 0 to ∞) (−1)ᵏ / (2k + 1)

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Domanda del libro di testo

55–70. More sequences

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


{(75n⁻¹ / 99ⁿ) + (5ⁿsinn / 8ⁿ)}

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Domanda del libro di testo

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


∑ (k = 1 to ∞) 1 / (2k − √k)

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