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
Ch. 10 - Sequences and Infinite Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 10, Problema 10.3.83
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)
Guida verificata passo dopo passo1
Recognize that the given series is a sum of two separate infinite series: \( \sum_{k=1}^\infty \left( \frac{1}{3} \times \left( \frac{5}{6} \right)^k + \frac{3}{5} \times \left( \frac{7}{9} \right)^k \right) = \sum_{k=1}^\infty \frac{1}{3} \left( \frac{5}{6} \right)^k + \sum_{k=1}^\infty \frac{3}{5} \left( \frac{7}{9} \right)^k \).
Identify each series as a geometric series of the form \( \sum_{k=1}^\infty ar^k \), where \(a\) is the coefficient and \(r\) is the common ratio. For the first series, \(a = \frac{1}{3}\) and \(r = \frac{5}{6}\). For the second series, \(a = \frac{3}{5}\) and \(r = \frac{7}{9}\).
Check the convergence of each geometric series by verifying if the absolute value of the common ratio \( |r| < 1 \). Since \( \frac{5}{6} < 1 \) and \( \frac{7}{9} < 1 \), both series converge.
Use the formula for the sum of an infinite geometric series starting at \(k=1\): \[ S = \frac{ar}{1 - r} \]. Apply this formula to each series separately.
Calculate the sum of the original series by adding the sums of the two geometric series: \[ S_{total} = \frac{\frac{1}{3} \times \frac{5}{6}}{1 - \frac{5}{6}} + \frac{\frac{3}{5} \times \frac{7}{9}}{1 - \frac{7}{9}} \].
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:
3mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Infinite Geometric Series
An infinite geometric series is a sum of terms where each term is a constant multiple (common ratio) of the previous one. It converges if the absolute value of the common ratio is less than 1, and its sum can be found using the formula S = a / (1 - r), where a is the first term and r is the common ratio.
Video consigliato:
Percorso guidato
06:00
Geometric Series
Convergence and Divergence of Series
A series converges if the sequence of its partial sums approaches a finite limit; otherwise, it diverges. For geometric series, convergence depends on the common ratio's magnitude. Understanding convergence is essential to determine whether the series sum exists or not.
Video consigliato:
Percorso guidato
06:52Convergence of an Infinite Series
Linearity of Series Summation
The sum of a series that is a sum of two series equals the sum of each series separately. This property allows breaking down complex series into simpler parts, evaluating each independently, and then combining the results to find the total sum.
Video consigliato:
Percorso guidato
07:17Linearization
Pratica correlata
Domanda del libro di testo
Suppose the sequence { aₙ} is defined by the explicit formula aₙ = 1/n, for n=1, 2, 3, .....Write out the first five terms of the sequence.
58
views
Domanda del libro di testo
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞)(cos(1 / k) – cos(1 / (k + 1)))
38
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
9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.
∑ (k = 1 to ∞) 1 / (2k − √k)
81
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