Skip to main content
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 10.8.53
Capitolo 10, Problema 10.8.53

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞)sin(1 / k⁹)

Guida verificata passo dopo passo
1
Identify the series given: \( \sum_{k=1}^{\infty} \sin\left(\frac{1}{k^9}\right) \). We want to determine if this series converges.
Recall that for very small angles \( x \), \( \sin x \approx x \). Since \( \frac{1}{k^9} \) becomes very small as \( k \to \infty \), we can approximate \( \sin\left(\frac{1}{k^9}\right) \approx \frac{1}{k^9} \) for large \( k \).
Compare the given series to the p-series \( \sum_{k=1}^{\infty} \frac{1}{k^p} \) where \( p = 9 \). We know that a p-series converges if \( p > 1 \). Since 9 is much greater than 1, the p-series \( \sum \frac{1}{k^9} \) converges.
Use the Limit Comparison Test to justify convergence: compute \( \lim_{k \to \infty} \frac{\sin\left(\frac{1}{k^9}\right)}{\frac{1}{k^9}} \). If this limit is a finite nonzero number, then both series behave similarly in terms of convergence.
Since the limit comparison test shows the given series behaves like a convergent p-series, conclude that \( \sum_{k=1}^{\infty} \sin\left(\frac{1}{k^9}\right) \) converges.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
6m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Convergence of Infinite Series

An infinite series converges if the sequence of its partial sums approaches a finite limit. Understanding convergence is essential to determine whether the sum of infinitely many terms results in a finite value or diverges to infinity or oscillates.
Video consigliato:
Percorso guidato
06:52
Convergence of an Infinite Series

Comparison Test for Series

The comparison test involves comparing the given series to a known benchmark series. If the terms of the given series are smaller than those of a convergent series, it also converges; if larger than a divergent series, it diverges. This test helps analyze series with complicated terms.
Video consigliato:
Percorso guidato
09:25
Direct Comparison Test

Behavior of sin(x) for Small x

For values of x close to zero, sin(x) is approximately equal to x. This approximation allows simplification of terms like sin(1/k⁹) to 1/k⁹ for large k, facilitating the use of p-series tests to determine convergence.
Video consigliato:
03:39
Integrals of Natural Exponential Functions (e^x)
Pratica correlata
Domanda del libro di testo

9–16. Divergence Test Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive.


∑ (k = 0 to ∞) k / (2k + 1)

65
views
Domanda del libro di testo

21–42. Geometric series Evaluate each geometric series or state that it diverges.  


27.1 + 1.01 + 1.01² + 1.01³ + ⋯

50
views
Domanda del libro di testo

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 ∞) k / √(k² + 1)

69
views
Domanda del libro di testo

25–26. Recursively defined sequences

The following sequences {aₙ} from n = 0 to ∞ are defined by a recurrence relation. Assume each sequence is monotonic and bounded.


b.Determine the limit of each sequence.


25.aₙ₊₁ = (1 / 2) aₙ + 8;a₀ = 80

42
views
Domanda del libro di testo

{Use of Tech} Error in a finite alternating sum

How many terms of the series ∑ (from k = 1 to ∞)(−1)ᵏ⁺¹ / k⁴ must be summed to ensure that the approximation error is less than 10⁻⁸?

62
views
Domanda del libro di testo

77–87. Absolute or conditional convergence

Determine whether the following series converge absolutely, converge conditionally, or diverge.

∑ (from k = 1 to ∞)(−1)ᵏ⁺¹(k² + 4) / (2k² + 1)

76
views