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.1g
Capitolo 10, Problema 10.R.1g

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.

Guida verificata passo dopo passo
1
Identify the general term of the series: \( a_k = \frac{k^2}{k^2 + 1} \).
Examine the behavior of \( a_k \) as \( k \to \infty \). Calculate the limit \( \lim_{k \to \infty} a_k = \lim_{k \to \infty} \frac{k^2}{k^2 + 1} \).
Since the degrees of numerator and denominator are the same, the limit is the ratio of the leading coefficients, which is \( 1 \). So, \( \lim_{k \to \infty} a_k = 1 \).
Recall the Divergence Test (also called the Test for Divergence): if \( \lim_{k \to \infty} a_k \neq 0 \), then the series \( \sum a_k \) diverges.
Because \( \lim_{k \to \infty} a_k = 1 \neq 0 \), the series \( \sum_{k=1}^\infty \frac{k^2}{k^2 + 1} \) does not converge; it 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.

Definition of Series Convergence

A series converges if the sequence of its partial sums approaches a finite limit as the number of terms goes to infinity. If the partial sums do not approach a finite value, the series diverges. Understanding this helps determine whether the given infinite sum converges or not.
Video consigliato:
Percorso guidato
06:52
Convergence of an Infinite Series

Term Test for Divergence

If the terms of a series do not approach zero as k approaches infinity, the series must diverge. This is a quick test to check convergence; if the limit of the general term is not zero, the series cannot converge.
Video consigliato:
Percorso guidato
05:44
Divergence Test (nth Term Test)

Behavior of the General Term (k² / (k² + 1))

Analyzing the limit of the term k² / (k² + 1) as k approaches infinity shows it approaches 1, not zero. Since the terms do not tend to zero, this indicates the series ∑ (k² / (k² + 1)) diverges by the term test.
Video consigliato:
Percorso guidato
06:11
Introduction to Riemann Sums
Pratica correlata
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 k = 1 to ∞)tanh(k)

76
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

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

60
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