Domanda del libro di testo
40–62. Choose your test Use the test of your choice to determine whether the following series converge.
∑ (k = 1 to ∞) 1 / k^(1 + p),p > 0
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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
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 10, Problema 10.3.79
72–86. Evaluating series Evaluate each series or state that it diverges.
∑ (k = 2 to ∞) ln((k + 1)k⁻¹) / (ln k × ln(k + 1))
Guida verificata passo dopo passo1
First, rewrite the general term of the series to simplify the expression inside the summation. The term is given by \(\frac{\ln\left((k+1)k^{-1}\right)}{\ln k \times \ln(k+1)}\). Use the logarithm property \(\ln(a/b) = \ln a - \ln b\) to rewrite the numerator as \(\ln(k+1) - \ln k\).
Substitute the simplified numerator back into the term to get \(\frac{\ln(k+1) - \ln k}{\ln k \times \ln(k+1)}\). Then, separate this fraction into two parts: \(\frac{\ln(k+1)}{\ln k \times \ln(k+1)} - \frac{\ln k}{\ln k \times \ln(k+1)}\).
Simplify each part of the separated fraction. The first part simplifies to \(\frac{1}{\ln k}\) and the second part simplifies to \(\frac{1}{\ln(k+1)}\). So the general term becomes \(\frac{1}{\ln k} - \frac{1}{\ln(k+1)}\).
Recognize that the series is telescoping because each term is of the form \(a_k - a_{k+1}\), where \(a_k = \frac{1}{\ln k}\). Write out the first few terms explicitly to see the cancellation pattern.
Use the telescoping property to express the partial sum \(S_n = \sum_{k=2}^n \left( \frac{1}{\ln k} - \frac{1}{\ln(k+1)} \right)\) as \(\frac{1}{\ln 2} - \frac{1}{\ln(n+1)}\). Then analyze the limit of \(S_n\) as \(n \to \infty\) to determine whether the series converges or diverges.
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Convergence and Divergence of Infinite Series
An infinite series converges if the sum of its terms approaches a finite limit as the number of terms grows indefinitely. Otherwise, it diverges. Determining convergence often involves applying tests that analyze the behavior of the terms or partial sums.
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Percorso guidato
06:52
Convergence of an Infinite Series
Properties of Logarithms
Logarithmic properties, such as ln(a/b) = ln(a) - ln(b) and ln(ab) = ln(a) + ln(b), help simplify complex expressions. Recognizing these can transform the series terms into simpler forms, making it easier to analyze or compare with known series.
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05:36Change of Base Property
Comparison and Limit Comparison Tests
These tests compare a given series to a known benchmark series to determine convergence. The limit comparison test uses the limit of the ratio of terms from two series; if the limit is finite and positive, both series share the same convergence behavior.
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Percorso guidato
07:45Limit Comparison Test
Pratica correlata
Domanda del libro di testo
48–63. Choose your test Determine whether the following series converge or diverge using the properties and tests introduced in Sections 10.3 and 10.4.
∑ (k = 1 to ∞) (5 / 6)⁻ᵏ
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Domanda del libro di testo
1–10. Choosing convergence tests Identify a convergence test for each series. If necessary, explain how to simplify or rewrite the series before applying the convergence test. You do not need to carry out the convergence test.
∑ (from k = 10 to ∞) 1 / (k − 9)⁵
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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 = 1⁄10ⁿ; n = 1, 2, 3, …
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Domanda del libro di testo
1–10. Choosing convergence tests Identify a convergence test for each series. If necessary, explain how to simplify or rewrite the series before applying the convergence test. You do not need to carry out the convergence test.
∑ (from k = 3 to ∞) (2k²) / (k² − k − 2)
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Domanda del libro di testo
6–9. Determine whether the following sequences converge or diverge, and state whether they are monotonic or whether they oscillate. Give the limit when the sequence converges.
{(−0.7)ⁿ}
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