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 10.5.25
Capitolo 10, Problema 10.5.25

9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.


∑ (k = 1 to ∞) 1 / (2k − √k)

Guida verificata passo dopo passo
1
Identify the given series: \( \sum_{k=1}^{\infty} \frac{1}{2k - \sqrt{k}} \). We want to determine if this series converges or diverges.
To apply the Comparison Test or Limit Comparison Test, find a simpler series to compare with. Notice that for large \(k\), \(2k - \sqrt{k} \approx 2k\), so the terms behave like \( \frac{1}{2k} \).
Consider the comparison series \( \sum_{k=1}^{\infty} \frac{1}{k} \), which is a harmonic series known to diverge.
Use the Limit Comparison Test by computing \( \lim_{k \to \infty} \frac{\frac{1}{2k - \sqrt{k}}}{\frac{1}{k}} = \lim_{k \to \infty} \frac{k}{2k - \sqrt{k}} \). Simplify this limit to determine if it is a finite positive number.
If the limit is a positive finite constant, then both series either both converge or both diverge. Since the harmonic series diverges, conclude the behavior of the original series accordingly.

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.

Comparison Test

The Comparison Test determines the convergence of a series by comparing it to a second series with known behavior. If the terms of the given series are smaller than those of a convergent series, it also converges. Conversely, if the terms are larger than those of a divergent series, it diverges.
Video consigliato:
Percorso guidato
09:25
Direct Comparison Test

Limit Comparison Test

The Limit Comparison Test compares two series by taking the limit of the ratio of their terms. If this limit is a positive finite number, both series either converge or diverge together. This test is useful when direct comparison is difficult but the terms behave similarly for large indices.
Video consigliato:
Percorso guidato
07:45
Limit Comparison Test

Behavior of Series Terms and Dominant Terms

Analyzing the dominant terms in the series' general term helps simplify the expression for large indices. For example, in 1/(2k - √k), the term 2k dominates √k as k grows large, so the series behaves like 1/(2k). Understanding this helps in choosing an appropriate comparison series.
Video consigliato:
Percorso guidato
05:44
Divergence Test (nth Term Test)
Pratica correlata
Domanda del libro di testo

9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.


∑ (k = 1 to ∞) (k² − 1) / (k³ + 4)

51
views
Domanda del libro di testo

72–86. Evaluating series Evaluate each series or state that it diverges.

∑ (k = 1 to ∞) ((1/3) × (5/6)ᵏ + (3/5) × (7/9)ᵏ)

53
views
Domanda del libro di testo

13–52. Limits of sequences

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


{√((1 + 1 / 2n)ⁿ)}

37
views
Domanda del libro di testo

33–38. {Use of Tech} Remainders in alternating series Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than 10⁻⁴ in magnitude. Although you do not need it, the exact value of the series is given in each case.


π / 4 = ∑ (k = 0 to ∞) (−1)ᵏ / (2k + 1)

61
views
Domanda del libro di testo

55–70. More sequences

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


{(75n⁻¹ / 99ⁿ) + (5ⁿsinn / 8ⁿ)}

59
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.

∑ (from k = 1 to ∞) ((-1)ᵏ⁺¹ × k²ᵏ) / (k! × k!)

58
views