Domanda del libro di testo
9–16. Divergence Test Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive.
∑ (k = 0 to ∞) k / (2k + 1)
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Ch. 10 - Sequences and Infinite Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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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
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Capitolo 10, Problema 10.5.3
What comparison series would you use with the Comparison Test to determine whether
∑ (k = 1 to ∞) 1 / (k² + 1) converges?
Guida verificata passo dopo passo1
Identify the given series: \( \sum_{k=1}^{\infty} \frac{1}{k^2 + 1} \). We want to determine if it converges using the Comparison Test.
Recall that for large values of \( k \), the term \( \frac{1}{k^2 + 1} \) behaves similarly to \( \frac{1}{k^2} \) because the \( +1 \) becomes insignificant compared to \( k^2 \).
Choose a comparison series that is simpler but related, such as \( \sum_{k=1}^{\infty} \frac{1}{k^2} \), which is a p-series with \( p = 2 > 1 \) and is known to converge.
Use the Comparison Test by noting that \( \frac{1}{k^2 + 1} < \frac{1}{k^2} \) for all \( k \geq 1 \), so if the larger series \( \sum \frac{1}{k^2} \) converges, then the original series also converges.
Conclude that the appropriate comparison series to use is \( \sum_{k=1}^{\infty} \frac{1}{k^2} \) because it helps establish convergence of the original series by the Comparison Test.
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Comparison Test
The Comparison Test determines the convergence or divergence of a series by comparing it to another 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
Choice of Comparison Series
To apply the Comparison Test effectively, select a simpler series whose convergence is known and whose terms closely resemble the original series. For ∑ 1/(k² + 1), the series ∑ 1/k² is a natural choice because for large k, 1/(k² + 1) behaves like 1/k².
Video consigliato:
Percorso guidato
06:00Geometric Series
Convergence of p-Series
A p-series ∑ 1/k^p converges if and only if p > 1. Since ∑ 1/k² is a p-series with p = 2, it converges. This fact helps conclude that ∑ 1/(k² + 1) also converges by comparison.
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Percorso guidato
04:30P-Series and Harmonic Series
Pratica correlata
Domanda del libro di testo
{Use of Tech} For what value of r does
∑ (k = 3 to ∞) r²ᵏ = 10?
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Domanda del libro di testo
38–39. Examining a series two ways Determine whether the following series converge using either the Comparison Test or the Limit Comparison Test. Then use another method to check your answer.
39. ∑ (k = 1 to ∞) 1 / (k² + 2k + 1)
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Domanda del libro di testo
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞)tan⁻¹(1 / √k)
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Domanda del libro di testo
21–42. Geometric series Evaluate each geometric series or state that it diverges.
27.1 + 1.01 + 1.01² + 1.01³ + ⋯
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Domanda del libro di testo
25–26. Recursively defined sequences
The following sequences {aₙ} from n = 0 to ∞ are defined by a recurrence relation. Assume each sequence is monotonic and bounded.
a.Find the first five terms a₀, a₁, ..., a₄ of each sequence.
25.aₙ₊₁ = (1 / 2) aₙ + 8;a₀ = 80
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