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 = 2 to ∞) √k / (ln¹⁰ k)
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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 10.6.7
Is it possible for a series of positive terms to converge conditionally? Explain.
Guida verificata passo dopo passo1
Recall the definitions: A series \( \sum a_n \) converges absolutely if \( \sum |a_n| \) converges, and it converges conditionally if \( \sum a_n \) converges but \( \sum |a_n| \) diverges.
Note that for a series of positive terms, each term \( a_n > 0 \), so \( |a_n| = a_n \). This means the series and its absolute value series are the same.
Since the series and its absolute value series are identical for positive terms, if the series converges, it must converge absolutely.
Therefore, a series with positive terms cannot converge conditionally because conditional convergence requires the absolute value series to diverge.
In summary, conditional convergence is only possible when the series has terms that are not all positive (i.e., some terms are negative or alternate in sign).
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Positive Term Series
A series with positive terms means each term is greater than zero. Such series are always non-negative and cannot have terms that cancel each other out, which affects the type of convergence they can exhibit.
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Percorso guidato
06:00
Geometric Series
Absolute Convergence
A series converges absolutely if the series of the absolute values of its terms converges. For positive term series, absolute convergence is equivalent to regular convergence since all terms are positive.
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Percorso guidato
07:51Choosing a Convergence Test
Conditional Convergence
Conditional convergence occurs when a series converges, but the series of absolute values diverges. This typically requires terms of varying signs, so a series with only positive terms cannot converge conditionally.
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Percorso guidato
07:51Choosing a Convergence Test
Pratica correlata
Domanda del libro di testo
Suppose the sequence { aₙ} is defined by the recurrence relation a₍ₙ₊₁₎ = n · aₙ , for n=1, 2, 3 ...., where a₁ = 1. Write out the first five terms of the sequence.
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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 ∞)1 / ln(eᵏ + 1)
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Domanda del libro di testo
21–42. Geometric series Evaluate each geometric series or state that it diverges.
39.∑ (k = 2 to ∞) (–0.15)ᵏ
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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 ∞) (2ᵏ / k⁹⁹)
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Domanda del libro di testo
35–44. Limits of sequences Write the terms a₁, a₂, a₃, and a₄ of the following sequences. If the sequence appears to converge, make a conjecture about its limit. If the sequence diverges, explain why.
aₙ = 3 + cos(π*ⁿ) ; n = 1, 2, 3, …
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