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 10.4.13
Capitolo 10, Problema 10.4.13

9–16. Divergence Test Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive.
∑ (k = 2 to ∞) k / ln k

Guida verificata passo dopo passo
1
Identify the general term of the series: \(a_k = \frac{k}{\ln k}\) for \(k \geq 2\).
Recall the Divergence Test (also known as the Test for Divergence): if \(\lim_{k \to \infty} a_k \neq 0\), then the series \(\sum a_k\) diverges.
Compute the limit of the general term as \(k\) approaches infinity: \(\lim_{k \to \infty} \frac{k}{\ln k}\).
Analyze the behavior of the limit: since \(k\) grows faster than \(\ln k\), the fraction \(\frac{k}{\ln k}\) grows without bound, so the limit does not approach zero.
Conclude that by the Divergence Test, since the limit of \(a_k\) is not zero, the series \(\sum_{k=2}^\infty \frac{k}{\ln k}\) 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:
2m

Concetti chiave

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

Divergence Test

The Divergence Test states that if the limit of the terms of a series does not approach zero as k approaches infinity, then the series diverges. It is a quick way to check divergence but cannot confirm convergence if the limit is zero.
Video consigliato:
Percorso guidato
05:44
Divergence Test (nth Term Test)

Behavior of the General Term

Analyzing the behavior of the general term k / ln(k) as k approaches infinity helps determine if the terms approach zero. Since ln(k) grows slower than k, the term k / ln(k) tends to infinity, indicating the terms do not approach zero.
Video consigliato:
Percorso guidato
05:44
Divergence Test (nth Term Test)

Infinite Series and Convergence

An infinite series converges only if the sum of its terms approaches a finite limit. If the terms do not approach zero, the series cannot converge. Understanding this principle is essential to apply tests like the Divergence Test correctly.
Video consigliato:
Percorso guidato
06:52
Convergence of an Infinite Series
Pratica correlata
Domanda del libro di testo

Property of divergent series Prove Property 2 of Theorem 10.8: If ∑ aₖ diverges, then ∑ caₖ also diverges, for any real number c ≠ 0.

69
views
Domanda del libro di testo

23–38. Divergence, Integral, and p-series Tests Use the Divergence Test, the Integral Test, or the p-series test to determine whether the following series converge.

∑ (k = 1 to ∞) 1 / ∛k

71
views
Domanda del libro di testo

55–70. More sequences

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


{nsin³(nπ / 2) / (n + 1)}"

47
views
Domanda del libro di testo

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.

1 + (1 / 2)² + (1 / 3)³ + (1 / 4)⁴ + ⋯

69
views
Domanda del libro di testo

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


∑ (from k = 1 to ∞)(ln²k) / k³ᐟ²

52
views
Domanda del libro di testo

What comparison series would you use with the Comparison Test to determine whether ∑ (k = 1 to ∞) 2ᵏ / (3ᵏ + 1) converges?

66
views