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.R.87
Capitolo 10, Problema 10.R.87

77–87. Absolute or conditional convergence
Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (from k = 1 to ∞)(−2)ᵏ⁺¹ / k²

Guida verificata passo dopo passo
1
Identify the given series: \( \sum_{k=1}^{\infty} \frac{(-2)^{k+1}}{k^2} \). Notice that the terms involve \((-2)^{k+1}\), which grows exponentially in magnitude, and the denominator \(k^2\), which grows polynomially.
To analyze convergence, first consider the absolute value of the terms: \( \left| \frac{(-2)^{k+1}}{k^2} \right| = \frac{2^{k+1}}{k^2} \). This transforms the series into \( \sum_{k=1}^{\infty} \frac{2^{k+1}}{k^2} \).
Check for absolute convergence by testing the series \( \sum_{k=1}^{\infty} \frac{2^{k+1}}{k^2} \). Since \(2^{k+1}\) grows exponentially and \(k^2\) grows polynomially, the terms do not approach zero fast enough for convergence. Use the root test or ratio test to confirm this behavior.
Next, consider conditional convergence by analyzing the original series with alternating signs. However, the factor \((-2)^{k+1}\) does not produce a simple alternating sign pattern because the base is \(-2\), not \(-1\). The terms actually grow in magnitude exponentially, so the series does not satisfy the conditions for the Alternating Series Test.
Conclude that since the series does not converge absolutely (step 3) and does not converge conditionally (step 4), the series diverges.

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:
5m

Concetti chiave

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

Absolute Convergence

A series ∑a_k converges absolutely if the series of absolute values ∑|a_k| converges. Absolute convergence guarantees convergence regardless of the sign of terms, and it implies the original series converges. Testing absolute convergence often involves comparison or p-series tests.
Video consigliato:
Percorso guidato
07:51
Choosing a Convergence Test

Conditional Convergence

A series converges conditionally if it converges, but does not converge absolutely. This means ∑a_k converges, but ∑|a_k| diverges. Conditional convergence often occurs in alternating series where the terms decrease in magnitude and approach zero.
Video consigliato:
Percorso guidato
07:51
Choosing a Convergence Test

Alternating Series Test

The Alternating Series Test states that an alternating series ∑(-1)^k b_k converges if the sequence b_k is positive, decreasing, and approaches zero. This test helps determine conditional convergence when absolute convergence fails.
Video consigliato:
Percorso guidato
10:54
Alternating Series Test
Pratica correlata
Domanda del libro di testo

Explain why or why not

Determine whether the following statements are true and give an explanation or counterexample.


g.The series ∑ (from k = 1 to ∞) (k² / (k² + 1)) converges.

39
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)ᵏk·e⁻ᵏ

67
views
Domanda del libro di testo

12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.

aₙ = (2ⁿ + 5ⁿ⁺¹) / 5ⁿ

39
views
Domanda del libro di testo

b.Does the series ∑ (from k = 1 to ∞) k/(k + 1) converge? Why or why not?

86
views
Domanda del libro di testo

42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.

∑ (from j = 0 to ∞)2 ⋅ 4ʲ / (2j + 1)!

35
views
Domanda del libro di testo

Building a tunnel — first scenario

A crew of workers is constructing a tunnel through a mountain. Understandably, the rate of construction decreases because rocks and earth must be removed a greater distance as the tunnel gets longer. Suppose each week the crew digs 0.95 of the distance it dug the previous week. In the first week, the crew constructed 100 m of tunnel.


b.What is the longest tunnel the crew can build at this rate?

41
views