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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 10.8.59
Capitolo 10, Problema 10.8.59

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞)k! / (kᵏ + 3)

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First, write down the general term of the series: \(a_k = \frac{k!}{k^k + 3}\).
To determine convergence, consider the behavior of \(a_k\) as \(k\) approaches infinity. Since \(k^k\) grows very rapidly, compare the growth rates of the numerator \(k!\) and the denominator \(k^k\).
Use the Ratio Test, which involves computing the limit \(L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right|\). Substitute \(a_k\) and simplify the expression:
\[L = \lim_{k \to \infty} \frac{(k+1)!}{(k+1)^{k+1} + 3} \cdot \frac{k^k + 3}{k!}.\]
Simplify the factorial and powers, then evaluate the limit \(L\). If \(L < 1\), the series converges absolutely; if \(L > 1\), it diverges; if \(L = 1\), the test is inconclusive and another test should be applied.

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Convergence of Infinite Series

An infinite series converges if the sequence of its partial sums approaches a finite limit. Determining convergence involves analyzing the behavior of the terms as the index grows large, ensuring the sum does not diverge to infinity or oscillate indefinitely.
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Percorso guidato
06:52
Convergence of an Infinite Series

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 especially useful for series involving factorials and exponential terms.
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Percorso guidato
05:35
Ratio Test

Factorials and Exponential Growth Comparison

Factorials (k!) grow faster than exponential functions like k^k for large k, but comparing their growth rates helps determine term behavior. Understanding how factorials compare to powers and exponentials is crucial for applying convergence tests effectively.
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Factorials
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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.

∑ (from k = 1 to ∞) (−1)ᵏ / √(k³ᐟ² + k)

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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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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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Domanda del libro di testo

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

∑ (from k = 3 to ∞) 5 / (2 + lnk)

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